Multibeam Satellite Frequency/Time Duality Study and Capacity Optimization

1 Multibeam Satelli te Frequenc y/T ime Duality Study and Capacity Optimization Jiang Lei, Stude nt Member , I EEE, M. A. V ´ azquez -Castro, Senior Member , I EEE, Abstract —In this paper , we in v estigate two new candid ate transmission schemes, Non-Orthogonal Fr equency Reuse (NOFR) and Beam-Hoping (BH). They operate in different domains (frequency and time/space, respectiv ely), and we want to know which domain shows ov erall best perfor mance. W e propose a nov el formulation of th e S ignal-to-Interference p lus Noise Ratio (SINR) which allows us to prov e the frequency/time duality of these schemes. Further , we propose two novel capacity optimization approaches assuming per-beam SINR constraints in order to use the satellite resources (e.g . power and bandwidth ) more efficiently . Moreov er , we develop a general methodology to include technological constraints due to realistic implementations, and obtain th e main factors th at prev ent the two techn ologies dual of each other in practice, and formulate the technological gap between them. The Sh annon capacity (upper bound) and current state-of-the-art coding and modul ations are an alyzed in order to q uantify the gap and to ev aluate the perf ormance of the two candidate schemes. Simulation results show significant impro vements in terms of power gain, spectral efficiency and traffic ma tching ratio when comparing with con v entional systems, which are designed based on uniform bandwidth and p ower allocation. The results also show that BH system t urns out to show a less complex design and p erf orms better than NOFR system sp ecially for non-real time services. Index T erms —Mu ltibeam Satellite, Duality , Beam-Hoppin g, Frequency-reuse, and Time-reuse. I . I N T RO D U C T I O N T HE efficient managem ent of mu ltibeam a ntenna re - sources, e.g. power , b andwidth a nd time- slot, is c rucial for eco nomic comp etiti veness. Specifically , in modern satellite networks, each satellite uses mu ltiple beams, each of whic h illuminates a cell on th e grou nd to serve a coverag e area. Multibeam antenn a tech nology is used bec ause it can inc rease the total system capacity significantly [ 1]–[3] . Howe ver, each beam will compete with others for resources to achieve satisfactory communica tion. This is d ue to the fact that the traffic demand thr ougho ut the coverage is potentially hig hly asymmetrical among the beam s. T herefor e, th e satellite re- quires a certain degree of flexibility in allo cating th e power , bandwidth and time-slot resources to ach iev e a g ood m atch between offered and requ ested traffic. There are som e preceden ts of re source allocation optimiza- tion techniques for satellite systems. In [4 ], [5], the auth ors in vestigate the dyn amic b andwidth allocation techniq ues, and in [6] the autho rs pr opose a quasi-optimal solution to manage the f requency slots allo cation to service providers, howe ver , the results are only fo r th e satellite up link. A p ower allocatio n Jiang Lei and M. A. V ´ azquez -Castro are with the Department of T elec om- municati ons and System Engineering , Univ ersita t Aut ´ onoma de Barcelona, Barcel ona 08193, Spain (e -mail: jiang .lei@uab .cat; angele s.va zquez@u ab .cat) . policy is proposed in [ 7], which sug gests to stabilize th e system based on the amoun t of unfinishe d work in the queu e and the chann el state, an d a routin g decision is m ade for the maximum to tal throug hput. Howe ver , the auth ors do n ot taking into account th e co- channel interferen ce. In [8], [ 9], a tradeoff strategy is propo sed betwe en different ob jectiv es an d system o ptimization. T he p ower allocation is optimized based on the traffic distribution an d ch annel cond itions. Howe ver , the co- channel in terference is not taken into account. In [1 0], a joint power and carrier a llocation problem is discussed, howe ver, it foc us o n the return (R TN) up link. In [1 1], [12] , the au thors fo cused on the capacity optim ization in multibea m satellite sy stem, and the d uality in frequ ency and time d omain is studied. Th e optimiza tion prob lem of power and car rier allocation has also b een add ressed in terrestrial ne tworks. E.g., the autho rs in [ 13] prop ose an axiomatic-based interferen ce model fo r Sign al-to-In terference plu s Noise (SINR) balancin g problem , but the conclu sions cannot be directly extrap olated to a satellite scen ario. Althoug h th e resourc e allocation op- timization has been stud y extensi vely , the ob jectiv e of this paper is d ifferent from the afor ementione d literatures in var- ious aspects. E.g., m ost of the existing resource allocation optimization approach es focus on terrestrial netw orks or on th e satellite R TN link , while we focu s on the satellite FWD link . In addition, we address the resource allo cation according to the realistic asymmetric traffic distribution by managin g the co- channel interfer ence due to frequency r euse. Existing work has focused on th e analysis of the resources alloca tion in frequency domain. Our aim is to ch aracterize th e best resour ce allocation scheme in multi-dom ain, and to show in which do main the overall perfo rmance is best. In this paper, we investigate two new transm ission schem es, Non-Orth ogonal Frequ ency Reu se (NOFR) techniqu e and Beam-Hopp ing (BH), which hav e been chosen a s candidates to replace current r egular fr equency reuse transmission scheme. The first one is d esigned based o n the fr equency d ivision over a flexible payload design which allows man aging interfer ence as an alternativ e to a complete orthogo nal f requen cy reuse. The secon d on e is based o n th e time/space division. Both technique s can potentially cope with the asymmetric traffic distribution as op posed to cur rent satellite resou rces allo cation scheme, which is designed to allocate fixed p ower and b and- width to each grou nd cell. This lead s to a waste o f resources in low traffic require ment beam s. On the contr ary , it does n ot satisfy traffic d emand in the ho t g round ce lls, wh ere the tr affic requirem ent is high. In this paper, we study the d uality between NOFR and BH, i.e. frequ ency and time/space duality . The concept of 2 duality gives rise to m any interesting p roperties to simplify system mod els. Gener ally speaking, a duality tran slates co n- cepts, theorems or mathem atical structures i nto other conc epts, theorems or structures, in a on e-to-on e fashion, o ften (but not always) by means of an inv olution operatio n: if the d ual of A is B , then the dual of B is A . E.g ., the d uality of space and time is studied in [14], [15], Gau ssian multiple- access/broadca st chan nel du ality is discussed in [16], uplink and downlink du ality is presen ted in [ 17], [18] . W e develop a general methodo logy to study the duality of th e two schemes that also considers the technolog ical constrain ts due to realistic implementatio ns, and obtain th e main factors th at p revent the two schemes be in practice dual of each other . The novel co ntributions o f this paper can be summarized as follows: • Th e frequen cy and time duality is f ormulated for the multibeam satellite system, and the d uality con ditions are derived for a practical system. • W e prove that ne w tr ansmission schemes, NOFR and BH, can match much better th an the con vention al design in the realistic asym metric traffic mod el, an d also prove that the BH system perfor ms only slightly better than NOFR. The rest of this pa per is organized as f ollows: In Sectio n II, the p roblem statement is p resented. I n Section III, we mo del the multibeam downlink system to obtain a m athematical expression of SINR. The duality of NOFR and BH is d iscussed in Section IV. In Sectio n V, we fo rmulate and so lve the satellite capacity optimization problems. In Section VI, the technolog ical gap is ob tained with a rea listic system p ayload model. T he simulation results are presen ted in Section VII. In Section VIII, we draw the conclusion s of the pap er . I I . P RO B L E M S TA T E M E N T In multibeam satellite systems, the be amformin g antenna generates K beams over the coverage ar ea. For both NOFR and BH systems, we firstly introd uce s ome payload param eters (as shown in T a ble I). • Gran ularity: B c is th e car rier g ranularity de fined as B c = B tot / N c in NO FR system. I t means th at the a llocated bandwidth p er g round cell shou ld be an integral multiple of B c . W e u se T s , with the same meaning b ut in BH system, i.e. the minim um u nit of time dur ation th at can be allocated per cell. • Resour ce allocatio n matrix: w ij and t ij are the elemen ts of the resource allocation matr ix for the NOFR and BH systems, respectiv ely . T he matr ix indica tes which carrier or time -slot j is allocated to the gro und cell i . Note th at BH ca n d irect the satellite be ams to specific g round cells, i.e. it is a space allocation too. In the ca se o f a NOFR scheme , each grou nd cell can be allocated a variable num ber of carriers (e. g. N i , as shown in Fig.1) depend ing on the traffic requir ement. Carrier s c an be re-used throughou t th e coverage, but we d o not impo se any restrictions on the f requency reuse, it will be gi ven by the resour ce optimization (i.e. interferen ce m inimization for a g i ven traffic demand pattern ) and the refore will be no n- orthog onal. In the case of a BH system, the total bandwidth is 1 2 N i B 1 B 2 B i B i+1 B K B tot B c Fig. 1. Bandwidth segmentati on. simultaneou sly used by a set o f the gr ound cells du ring a time- slot ( T s ). W e assum e that the resour ce allocation takes place during a g iv en time window d i vided into N t time-slot. Each groun d cell can be allocated a variable number of time-slot. Note that b oth techniq ues allow a number of g round cells to use the same frequen cy band or tim e-slot, resulting in co- channel in terference . The problem tackled in this pape r is to optimize the capacity by takin g into ac count the co-chann el interferen ce. Fu rther, we prove th e duality o f th ese techn iques by developing a for mulation that also allows including techn o- logical constraints. More over , we compare the perf ormance of the p roposed new transmission schemes with the curren t one for the re alistic asymmetric tr affic mod el. T o do so, we propo se a novel iterative alg orithm, which do n ot only optimize the power an d ban dwidth allocation (fo r NOFR systems), but also optimize the structur e of the spectral mask matrix W . This ma trix indicates which ca rriers are allocated p er-beam in order to minimize the co-channe l interfer ence. Althou gh the power and carrier optimizatio n pr oblem h as been addr essed in terrestrial networks, it is new in satellite comm unications. I I I . M U LT I B E A M S Y S T E M M O D E L In this section, we first formu late the multibeam system model in fre quency do main (i.e. fo r the NOFR scheme ). Af ter that we state the conditio ns for du ality and prove that NOFR and BH are dual of ea ch oth er and hence, the fo rmulation is also valid in tim e dom ain(i.e. for the BH scheme) . This d ual formu lation allows us to deriv e a unique SINR expr ession, which will be used in the following section f or cap acity opti- mization. Follo wing, we in troduce the different sub-m odels. A. Chan nel Mod el W e d o an an alysis in time an d hence the c hannel attenuatio n correspo nds to the free space losses an d atmosph eric losses (in case of f requenc ies above Ka band) . W e assume an instanta- neous analy sis with fixed coefficient. The channel attenu ation amplitude matrix A ∈ C K × K is defined as A = d iag { α 1 , α 2 , · · · , α K } , (1) where α i denotes the channel attenuation factor over th e destination user beam i . B. Anten na Mo del An Array Feed Reflector (AFR) based Anten na system is assumed, it can generate a regular beam grid ar ray con sisting of a very hig h n umber o f highly overlapping, narrow beam width, compo site u ser beams. Each bea m is synthesized by adding array elemen ts whose phases and amp litudes are ad- justable, and hence we ca n provide flexible power allo cation 3 by controllin g the On -Board Processor (OBP). Th erefore, we suppose that the antenn a gain matrix G ∈ C K × K is giv en as G =       g 11 g 12 · · · g 1 K g 21 g 22 · · · g 2 K . . . . . . . . . . . . g K 1 g K 2 · · · g K K       , (2) where | g ij | 2 ∈ R 1 × 1 is the antenna gain of the o n-boar d antenna feeds for the j th beam tow ards the i th user beam. C. Rece ived Sig nal Model In the frequ ency dom ain, the tran smitted symbols over N c carriers to beam i ( i = 1 , 2 , · · · , K ) are defined as x i = [ x i 1 , x i 2 , · · · , x iN c ] T . Let the spectral m ask matrix W ∈ R N c × K be d efined as W = [ w 1 , w 2 , · · · , w K ] , and the i th column vector w i ∈ R N c × 1 be de fined as w i = [ w i 1 , w i 2 , · · · , w iN c ] T , which is the spec tral mask vector for beam i a nd indicates which TDM carriers and how much power is allocated to beam i . Let H = AG b e the overall ch annel matrix, a nd W i = diag { w i } . Then the received signal by a ll the N c carriers for beam i , y i ∈ C N c × 1 , can b e expressed a s d esired signal and interferen ce as y i = h ii ˜ x i + K X k =1( k 6 = i ) h ik ˜ x k + n i , (3) where ˜ x i is the spectral masked symbol vector for beam i , defined as ˜ x i = W i x i . T he fir st term correspon ds to th e desired sign als co ming from the i th on-board antenn a. The second term is the sum of interferen ce signals from the o ther on-bo ard a ntennas. n i ∈ C N c × 1 is a colum n vector of zero- mean complex circular Gaussian noise with variance σ 2 at beam i . D. Sign al-to-In terfer enc e plu s Noise Ratio In the freque ncy do main, the bandwidth is se gmented as shown in Fig.1. W e assume that th e whole ban dwidth is segmented in N c carriers. The spectral mask matrix can be reform ulated as W = [ ˜ w T 1 , ˜ w T 2 , · · · , ˜ w T N c ] T , where ˜ w j = [ w 1 j , w 2 j , · · · , w K j ] , indicates which beams are allocated carrier j . Let th e i th row o f H be de fined as h i = [ h i 1 , h i 2 , · · · , h iK ] an d ˜ h i = h i | ( h ii =0) is the chann el of interferen ce contribution. W e assume that the amplitude of the transmitted symbols is n ormalized (i.e. | x ij | 2 = 1 , ∀ i = 1 , · · · , K ; ∀ j = 1 , · · · , N c ). Then, the tra nsmitted signal p ower of all the carrier s f or beam i can be given by the d iagonal elemen ts o f the ma trix S f i ∈ R N c × N c as (note that th e su perscript f an d t ind icate the expression in f requency an d time d omain, respectively) S f i = | h ii | 2 W i W H i . (4) And the co-chan nel in terference power of all the carr iers for b eam i can a lso b e gi ven by the diagonal elem ents o f the matrix U f i ∈ R N c × N c as U f i = d iag n  ˜ h i ˜ w H j ˜ w j ˜ h H i  j =1 , 2 , ··· ,N c o . (5) T ABLE I F R E Q U E N C Y - T I M E D U A L I T Y Frequenc y domain Time domain Granulari ty B c T s T otal number of carriers/ time-slot N c N t Resource allocati on matrix w ij t ij SINR ( γ ij ) γ f ij γ t ij Spectra l efficie ncy ( η ij ) η f ij η t ij Throughput for beam i R f i R t i Thus, the inter ference power plus the noise matrix , V f i , will be giv en as V f i = U f i + σ 2 I N c . (6) Consequently , the SINR for b eam i , defined as Γ f i ∈ R N c × N c , can be expressed as Γ f i = S f i ( V f i ) − 1 . (7) Obviously , Γ f i is a diag onal m atrix, b ecause both S f i and V f i are diag onal matrixes. Thu s, the SINR for the j th car rier used by beam i will be the j th diagonal elemen t of the matrix Γ f i . T his means that fo r each carrier j of beam i , the SINR can be formulated as γ f ij = | h ii w ij | 2 K X k =1( k 6 = i ) | h ik w kj | 2 + σ 2 . (8) For the R TN uplin k scenario, the auth ors in [10] fo rmulate the SINR in a similar way for a specific termina l. Howe ver , for the FWD downlink, we formulate the SINR per beam, and all th e carrier s’ SINRs are integrated in an equation (7), the SINR for a specific carrier (8) is also derived from (7). I V . F R E Q U E N C Y / T I M E D UA L I T Y In the previous section, expression (8) gi ves th e SINR in terms of the spectral mask vector (i.e., in fr equency domain ). In th is section, we will prop ose the frequen cy/time du ality of (8). For d oing so, we first state the dual expression in time domain. After th at, we can find the condition s for th e duality . A. Dua l S ystem Mod el In the time do main, the time window is segmented into N t time-slot. The time-slot mask matrix can be for mulated as T = [ ˜ t T 1 , ˜ t T 2 , · · · , ˜ t T N t ] T , where ˜ t j = [ t 1 j , t 2 j , · · · , t K j ] , ind i- cates which beam s ar e allocated time- slot j . Then, acco rding to the duality e lements in T ab le I, the tr ansmitted signa l power matrix S t i , the co-ch annel interferen ce power matrix U t i , the interferen ce power plu s the noise matrix V t i and the SINR matrix Γ t i in time dom ain can be for mulated as follows S t i = | h ii | 2 T i T H i , (9) U t i = d iag n  ˜ h i ˜ t H j ˜ t j ˜ h H i  j =1 , 2 , ··· ,N t o , (10) 4 V t i = U t i + σ 2 I N t , (11) Γ t i = S t i ( V t i ) − 1 . (12) The SINR of be am i an d time-slot j will be the j th d iagonal element of the m atrix Γ t i . Hen ce, the SINR can be formu lated as γ t ij = | h ii t ij | 2 K X k =1( k 6 = i ) | h ik t kj | 2 + σ 2 . (13) From a theo retical p oint of view [14] , (8) and (13) are d ual of each other . However , fo r a practical system, we d eriv e the duality conditio ns in the next section. B. Dua lity Con ditions From (8 ) and (13 ) we can extract the duality con ditions. In ord er to do so, we first e xpress the beam -lev el sum- rate throug hput as follows R f i = N c X j =1 B tot N c η f ij , (14) and the dual is R t i = N t X j =1 B tot N t η t ij , (1 5) where η ij = f ( γ ij ) is the spectral efficiency ( η ij can be η f ij or η t ij in fr equency d omain or time dom ain, respectively), and f ( γ ij ) equ als to log 2 (1 + γ ij ) for Shan non limit with Ga ussian coding, or can be a quasi-lin ear f unction in DVB-S2 [1 9] with respect to SINR . Hence, in o rder to obtain the duality cond itions, we assume that R f i = R t i and the throu ghpu t rate p er-carrier in freq uency domain (or p er time-slot in time dom ain ) is equiv alen t for each illumin ated beam. Th e following condition s should be fulfilled for systems to be dual in pr actice: • Gran ularity in frequen cy and time doma ins shou ld b e the same: N c = N t . • Th e entries of resource allocation matr ix should be the same in frequen cy a nd time domain s: w ij = t ij . • Th e spectral efficiency functio n f ( · ) should be the same for NOFR and BH systems in f requen cy and time do- mains, respectively: f ( η f ij ) = f ( η t ij ) . V . C A P AC I T Y O P T I M I Z AT I O N In th is section, we propo se two capacity optimization pr ob- lems under the constrain ts of the traffic requested p er-beam and overall av ailable power . The first pro blem, P1: capacity optimizing with co-chann el interf erence, maximizes the total capacity allocated with re spect to the traffic requ ested. Since the prob lem is n ot only non-co n vex but also to th e n eed of preservin g the geo metry of the matrix W . Therefo re, we propo se an iterativ e algorithm solu tion. The secon d one, P2: capacity o ptimizing with out co -chann el inter ference, is a sim- plified problem by assuming that the co-channel interference is negligible, hen ce, th e pr oblem can b e solved stra ightforwardly by the lagran gian appr oach. T ABLE II A L G O R I T H M S O L U T I O N F O R N O F R S Y S T E M 1: Initialize: R k ⇐ 0 , ∀ k . n it ⇐ 0 . W ⇐ 0 2: i ⇐ 0 . Generating beam set A s : A s = n i 1 , i 2 , · · · , i N | 0 ≤ R i n ˆ R i n ≤ R i n − 1 ˆ R i n − 1 < 1 o . where i n ∈ { 1 , 2 , · · · , K } , n = 1 , 2 , · · · , N . 3: n it ⇐ n it + 1 Repeat : i ⇐ i + 1 . k ⇐ A s ( i ) 4: Solve the Rayleigh quotient problem: arg max e H j S f k e j e H j V f k e j 5: w kj ⇐ e H j e j ( P sat ) 1 / 2 6: Update S f k , U f k . V f k ⇐ U f k + σ 2 I 7: go to step 3, until k > i N . 8: Update γ f kj , ∀ k , j . R k ⇐ P N c j =1 B tot N c η f ij , ∀ k . 9: go to step 2, until A s is empty or P K i =1 w H i w i ≤ P tot . A. P1: cap acity op timizing with co-channel interfer ence Obviously , γ f ij in for mula (8) n ot only dep ends o n the spectral mask vector of beam i ( w i ), but also depends on that of the co-ch annel b eams. And hence, the spectral mask vector for e ach beam m ust be op timized jointly with the others. The specific design o f on e beam’ s spectral mask vecto r may affect the cro sstalk experien ced b y o ther beam s. Hence it’ s a co mplicated task to design the spectral m ask m atrix W jointly . I n or der to best m atch the offered and th e r equested traffic pe r-beam, we develop a methodolo gy to solve the spectral m ask m atrix W in th is sectio n an d to jointly o ptimize power and carrier allocation . No te that we only discu ss the capacity optimization for NOFR system because BH is dual with NOFR, th us the formulation is also ap plicable fo r BH system by chang ing the duality param eters in T able I. Existing results in the references [ 20]–[ 23] is exclusively over th e power allocatio n. Howe ver , we assume an add itional degree of freedom: car rier allo cation (ban dwidth granu larity). W e pro pose to use binary power allo cation ( BP A) ( | w ij | 2 = { 0 , P sat } , i = 1 , 2 , · · · , K ; j = 1 , 2 , · · · , N c ) and quantized bandwidth allocation in ord er to decrease the complexity , where P sat is the TWT A saturation power pe r carrier . 1) Optimization Pr o blem F ormulatio n: The capacity opti- mization problem can be formu lated as max W K X i =1 R i ( W ) ˆ R i , subject to R i ≤ ˆ R i , (16) K X i =1 w H i w i ≤ P tot ; and | w ij | 2 = { 0 , P sat } , ∀ i, j. where ˆ R i is th e traffic re quested by b eam i , R i ( W ) is d efined in T ab le I. P tot is to tal av ailable satellite p ower , P sat is the saturation power per carrier, which is limited by the satellite amplifier . 2) Iterative Algo rithm Solution: The general an alytical so- lution of (16 ) is a comp lex p roblem d ue no t only to th e clea r non-co n vexity but also to the need of p reserving the geo metry 5 of the o ptimization model ( i.e. the structure of matrix W ). Therefo re, we p ropose an iterative algorithm solution, which is sum marized in T able II. The beam set A s is co nstituted by all the b eams, in which the traffic requ irement is not achieved (i.e. R k ˆ R k < 1 ). Quantities associated with the n th iteration are denoted b y n it . Each iteration is based o n a two-step proce ss. Firstly , we optimize sub space-by- subspace and ob tain an analytical solutio n to the sub-p roblem of allocatin g the carrier on a per-beam b asis (a s shown in step 4 of T able II) . The optimal carrier allocation p er-beam can be formulated as a generalized Rayleigh quotient, e.g. for beam i , the pr oblem can be form ulated as: arg max j e H j S f i e j e H j V f i e j , subject to K X i =1 w H i w i ≤ P tot . (17) where e j ∈ R N c × 1 is stan dard b asis vector, which d enotes the vector with a “ 1” in the j th coordin ate and 0’ s elsewhere. The solution of the genera lized Rayleigh quotien t p roblem shown in (17) is given as e j = υ max ( S f i ( V f i ) − 1 ) = υ max ( Γ f i ) , (18) where υ max ( Γ f i ) (as expressed in 7) indicates th e eigenv ector related to the maximu m eigenv alue of matrix Γ f i . Secondly , we obtain the power allocated to the selected carriers fro m the p ower constraint (as shown in step 5 of T able II). w ij for the j th carrier of beam k can be o btained with the solution of e j as w ij = e H j e j ( P sat ) 1 / 2 . (19) After each iteration, we up date matrix S f i and V f i accordin g to the updated spectral mask matrix W . B. P2: cap acity op timizing without co-channel in terfer ence In this section, we assume that the c o-chann el interf erence is negligible, b ecause th e co-chan nel b eams (in frequen cy do - main) or the simultaneously illu minated beams (in time/sp ace domain) can be separated far fro m each o ther in practice. In this way , the capacity o ptimization can be red uced to a co n vex problem . T wo c ost fun ctions are pro posed to solve the frequ ency and time/space capacity op timizing problem withou t co-channe l interferen ce. No te that we only discu ss the optimization pr ob- lem fo r BH sy stem b ecause NOFR is d ual with BH (see Section IV), thus th e fo rmulation is also applicable for NOFR system b y ch anging the r elated p arameters (e.g. T s → B c , N t → N c ). 1) n -th Order Differ en ce Cost Functio n: Here we want to match allocated bit r ate R i to req uested b it rate ˆ R i as closely as po ssible, i.e. , we want to min imize a gen eral function of the difference between { R i } and { ˆ R i } across all the g round cells. If a n n - th o rder deviation co st f unction is used, the problem can be formulated as min K X i =1    R i − ˆ R i    n , subject to R i , ≤ ˆ R i (20) K X i =1 N t i ≤ N re max N t . N re max is the numb er of cells illumin ated simu ltaneously , w hich is a satellite p ayload c onstraint. N t i is the numb er of time-slo t allocated to groun d cell i . W e a ssume that the power allocated to each time-slot is con stant. And h ence R i can be simplified as (assuming Gaussian codes): R i = N t i N t B tot log 2 (1 + γ i ) . (21) Since the co-chann el inter ference is assumed to be neg- ligible, the optimization pro blem shown in (20) is conv ex. Therefo re, the lagrang ian fu nction is g iv en as J ( N t i ) = K X i =1   R i − ˆ R i   n + λ K X i =1 N t i − ! . (22) Let ∂ J ( N t i ) ∂ N t i = 0 , w e can obtain N t i = ˆ R i N t B tot log 2 (1 + γ i ) −  λ n  1 n − 1  N t B tot log 2 (1 + γ i )  n n − 1 , (23) where λ is the la grange multiplier an d determ ined f rom the total av ailab le time-slot con straint, which can be ob tained by solving the equation K X i =1 N t i = N re max N t . (24) From (23) and (24) we can obtain λ = n       K X i =1 ˆ R i N t B tot log 2 (1 + γ i ) − N re max N t K X i =1  N t B tot log 2 (1 + γ i )  n n − 1       n − 1 . (25) If we replace λ in (23) with (25 ), the so lution will b e N t i = ˆ R i N t B tot log 2 (1 + γ i ) − K X k =1  ˆ R k N t B tot log 2 (1 + γ k )  − N re max N t K X k =1  log 2 (1 + γ i ) log 2 (1 + γ k )  n n − 1 . (26) W ith the the nu mber of time- slot allocated to eac h gr ound cell ( N t i ), th e throu ghput alloc ated to each cell ( R i ) can be calculated with (21 ). W e should note tha t the solution in (26 ) is ind ependen t of th e orde r n ( n ≥ 2 ) in our case, since we suppose BP A, n o co-channel inter ference and th e s ame channel attenuation factor ( α i ) for all the grou nd cell. 6 2) F airness Cost Function: Another way to match allocated capacity R i to req uested capacity ˆ R i is to m aximize the ratio between them as max K Y i =1  R i ˆ R i  ω i , subject to R i ≤ ˆ R i , (27) K X i =1 N t i ≤ N re max N t , where ω i is the weighting factor tha t repr esents th e pr iority of each b eam. The p roblem (27) can be easily converted to a conv ex pro blem by introducin g th e logar ithm in the ob jectiv e function . Thu s, the op timization problem is conv erted to max K X i =1 ω i log 2  R i ˆ R i  . (28) Thus, the lagrangian functio n is given as J ( N t i ) = − K X i =1 ω i log 2  R i ˆ R i  + λ K X i =1 N t i − N re max N t ! . (29) Let ∂ J ( N t i ) ∂ N t i = 0 , th en N t i = ω i ˆ R i N t λ ln 2 B tot log 2 (1 + γ i ) . (3 0) W ith given co nstraint K X i =1 N t i = N re max N t , the lagrange multiplier can be solved as λ = K X i =1 ω i ˆ R i N t B tot log 2 (1+ γ i ) N re max N t ln 2 . (31) The solution will be (replace λ in (30 ) with (3 1)) N t i = ω i ˆ R i N t log 2 (1 + γ i ) N re max N t K X k =1 ω k ˆ R k N t log 2 (1 + γ k ) . (32) Therefo re, the throu ghpu t allocated to each gr ound cell ( R i ) can be calculated with (21) . V I . T E C H N O L O G I C A L G A P From Section IV -B we can see th at the spectral efficiency that eac h tech nology can provide makes the real difference. Therefo re, NO FR and BH systems are not comp letely dual of each othe r . In th is section, we will demon strate the tec hnolog - ical gap between NOFR and BH. Note that we only consider the f orward (FWD) downlink, b ecause the FWD uplin k is not a big issue since power at the gateway can be greatly increased to compe nsate the attenuatio n. Equiv alent isotrop ically radiated power (EI RP) is define d as (in dB) E I RP = P sat − O B O − L repeater − L antenna + G tx , (33) where Output BackOff (OBO) is th e ra tio of maximu m o utput (saturation) p ower to actual outpu t power , L repeater is th e repeater loss, L antenna is the antenna f eed loss, a nd G tx is the satellite Tx. an tenna g ain. With known EIRP , we ca n ob tain FWD downlink C / N 0 (in dBHz) and SNR (in dB) as C / N 0 = E I RP − L propagation + ( G/T ) gt − 10 log 10 ( k B ) , (34) S N R = C / N 0 − 10 log 10 ( B c ) , (35) where L propagation is the p ropaga tion loss, ( G/T ) gt is th e groun d terminal G/T and k B is the Boltzman n co nstant. Let a = P sat − L repeater − L antenna + G tx − L propagation + ( G/T ) gt − 10 log 10 ( k B ) − 10 lo g 10 ( B c ) , a nd let x 1 and x 2 be th e OBO for NOFR an d BH systems, respectively . Theref ore, ( 35) can be reformu lated as S N R f = a − x 1 , or S N R t = a − x 2 . (36) Let the FWD downlink signa l to c o-chann el interfer ence SIR be given as y , There fore, the FWD downlink SINR can be formulated as S I N R − 1 down = S I R − 1 + S N R − 1 = y − 1 + 1 0 − ( a − x 10 ) , ( 37) where x can be x 1 or x 2 and S N R can be S N R f or S N R t for NOFR or BH system. Let the FWD uplink SINR be z , then the FWD who le link SINR is giv en as S I N R − 1 tot = S I N R − 1 up + S I N R − 1 down = z − 1 + y − 1 + 1 0 x − a 10 . (38) Let the who le FWD link SIN R be γ = S I N R tot , the spectral e fficienc y in the c ase o f Sh annon limit with Gaussian coding can be given as η = log 2 (1 + γ ) ≃ log 2 ( γ ) = − log 2 ( z − 1 + y − 1 + 1 0 x − a 10 ) , (39) where w e m ake a high SI NR ap prox given as, log 2 (1 + γ ) ≃ log 2 ( γ ) . Therefor e, the spectral efficienc y for NOFR and BH system are η f = − log 2 ( z − 1 + y − 1 + 1 0 x 1 − a 10 ) , (40) η t = − log 2 ( z − 1 + y − 1 + 1 0 x 2 − a 10 ) . (41 ) Let the tech nological gap of spectral efficiency be tween BH and NOFR system ∆ η be given as ∆ η = η t − η f = log 2 z − 1 + y − 1 + 1 0 x 1 − a 10 z − 1 + y − 1 + 1 0 x 2 − a 10 . (42) Let z , x 1 and x 2 be constant a nd x 1 > x 2 , ∆ η will be a monoto nically increasing func tion o f y . Ther efore, th e upper bound (maximum ) of the techno logical g ap ∆ η will be ∆ η max = ∆ η | y → + ∞ = lo g 2 1 + z 10 − ( a − x 1 10 ) 1 + z 10 − ( a − x 2 10 ) . (43) As we in dicated b efore, th e uplink is not r elev ant. Thus we can sup pose that th e up link SINR z is constant. T he resu lt of the technolog ical gap is demonstra ted in Fig.8, it is meaning ful for us to evaluate the perfo rmance of NOFR and BH, a nd to predict the tech nological gap be tween NOFR an d BH systems. 7 V I I . N U M E R I C A L R E S U LT S The objective of the simulatio n is: Firstly , to evaluate the per formanc e of the pro posed capac ity optimization ap - proach es. Seco ndly , to compare th e prop osed system d esign with th e conventional design, which is regula r frequency reuse ( f R = 7 ) an d un iform p ower allocation. Thirdly , to obtain the technolog ical g ap for a rea listic imp lementation. The p ayload parameters are defined in [24] . In order to fairly compare th e per forman ce with d ifferent number of b eams in th e same coverage ( e.g. the Eu ropean countries), we assume that the total traffic re quiremen t is the same fo r all the cases. The linear tr affic d istribution model is defined as ˆ R k = k β ; k = 1 , 2 , · · · , K , β is slope of the linear fun ction. The following par ameters are assume d in the simulations: P sat = 4 W att, B tot = 500 M Hz, N c = 11 2 , each cluster contain s 7 beam s ( as sh own in Fig.2), β = 8 × 10 6 bps for K = 1 2 1 , an d B c = B tot / N c = 4 . 46 4 3 MHz. The parameter s o f power ga in ( g ), spectral efficiency ( η ) and traffic matching ratio ( ρ ) ar e studied in the simulation, which are defined as the following. A. P erformance P arameters De finition 1) P ower Ga in: W e compare the amount of total power consump tion for joint power and bandwidth optimized allo- cation with that for unifor m power and bandwid th allocation when bo th ac hiev e the same u seful th rough put u sing the sam e total bandw idth. W e define the power gain g p as g p = K P uni K X k =1 w H k w k , (44) where P uni denotes the power per-beam of the unif orm allo- cation scheme. 2) Sp ectral Efficienc y: Th e spectral efficiency is defin ed based on the total allocated traffic a nd total allocated ban d- width as η = K X k =1 R k K X k =1 B k . (45) 3) T raffic Matching Ratio : In ord er to describe the satis- faction degree o f the allocated traffic with respect to the total request traffic, the traffic matching ratio is d efined here as ρ = K X k =1 R k K X k =1 ˆ R k . (46) B. Beam Layo ut a nd Antenna Model W e assume a gen eral beam layout mo del (shown in Fig.2). A fixed-size space is used to generate different number of beams, thus, the bea mwidth is decr easing as the numbe r of beams incre ases. It means th at the larger the nu mber of beams, : Cluster : Beam Coverage (Cell) Fig. 2. Beam layout. 0 5 10 15 20 25 30 35 40 0 1 2 3 4 5 6 x 10 10 N iteration Throughput [bps] K=49 K=121 K=225 Total Throughput Request Fig. 3. Conv ergence speed. 50 100 150 200 250 300 0 1 2 3 4 5 6 7 8 K [Number of Beams] Power Gain [dB] Shannon DVB−S2 2.5dB Fig. 4. Power gain ( g p ) vs. number of beams ( K ). the narrower the beam width. W e assume a tap ered aperture antenna mod el with 47 .14 dBi maximu m antenna gain . Then the SINR can be calculated in each itera tion of the algo rithm with a given link budget of a typ ical Ka- Band (19.9 5 GHz) satellite payload. C. Simulatio n Results In orde r to evaluate the r elev anc e of our iterati ve algor ithm (shown in T able II ), we perfor m a study of co n vergence. It can be observed fr om Fig.3 that the alg orithm is conv ergent for different nu mber o f beams, and the conv ergence is faster with the num ber of beam s increasing, e.g . our algo rithm runs 2 4 and 33 iterations for numb er o f 225 and 49 beams respectively . 8 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 2 2.5 3 3.5 4 4.5 5 K [Number of Beams] Spectral Efficiency ( η ) Shannon case (P1: capacity optimizing with co−channel interference) DVB−S2 case (P1: capacity optimizing with co−channel interference) Shannon case (uniform resource allocation) DVB−S2 case (uniform resource allocation) 1 bit/s/Hz 0.7 bit/s/Hz Fig. 5. Spectral efficie ncy ( η ) vs. number of beams ( K ). 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K [Number of beams] Traffic matching ratio ( ρ ) Shannon case (P1: capacity optimizing with co−channel interference) DVB−S2 case (P1: capacity optimizing with co−channel interference) Shannon case (uniform resource allocation) DVB−S2 case (uniform resource allocation) 15% Fig. 6. Traf fic matching ration ( ρ ) vs. number of beams ( K ). The reason is that the alg orithm allocates resources to all unsatisfied b eams in each iteration, th us, m ore tr affic will be allocated with larger number of beams. Conseq uently , all the beams will reach th e traffic requir ement faster . The algor ithm has been applied in the r ealistic satellite payload model an d proved in [24] that it is applicab le to the curren t multibe am satellite system. The po wer ga in with respect to the num ber of beams is shown in Fig.4. W e can see that about 6dB and 3.5dB power gain can be achieved by ca pacity optimizing of P1 with Gaussian codin g and DVB -S2 ModCod s (d efined in [ 19]), respectively (whe n K = 200 ). By optimizing the ca pacity achieved per-beam, we do not only r educe the p ower and band- width consumption o f small traffic requirement b eams, b ut also achieve reasona ble propor tional fairness fro m the viewpoint of user b eams. In Fig.5 , the result shows that the spectral efficiency decreases with the num ber o f beams increasing, especially when K > 20 0 . T he reason is that co -chann el interferen ce will increa se with the b eamwidth decreasing. In Fig.6 we can observe that the traffic matches better in case of larger number of be ams. Howe ver, the power consump tion an d the complexity will increase with larger number of b eams. Therefo re, we should balance the total achieved through put with resp ect to both power consumptio n and complexity . Fig.7 shows the traffic matchin g ratio with respect to different traffic distribution slop e. Obviously , the traffic matc hing ratio drops down with th e slope increasing. 1 3 5 7 9 11 13 15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 slope ( β ) [ × 10 6 bps] Traffic matching ratio ( ρ ) Shannon case (P1: capacity optimizing with co−channel interference) DVB−S2 case (P1: capacity optimizing with co−channel interference) Shannon case (uniform resource allocation) DVB−S2 case (uniform resource allocation) Fig. 7. Traf fic matching ration ( ρ ) vs. traffic slope ( β ). 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 ∆ OBO =x 1 −x 2 [dB] ∆η max [bits/s/Hz] ∆η max (x 2 =1dB) ∆η max (x 2 =3dB) ∆η max (x 2 =5dB) Fig. 8. ∆ OBO vs. ∆ η max . 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 x 10 8 Beam No# Throughput [bps] Linear Traffic Request per Beam with β =3 × 10 7 P2: Traffic Allocated with n−order (n=2) Difference Cost Function (BH) P2: Traffic Allocated with Fairness Cost Function (BH) P2: Traffic Allocated with n−order (n=2) Difference Cost Function (NOFR) P2: Traffic Allocated with Fairness Cost Function (NOFR) Fig. 9. Comparison of cost functions in terms of throughput. Because the traffic distribution is more asymmetric w ith larger slope. W e can see that the capacity optimiza tion can achieve better matching r atio compa red to the conventional d esign for both Shannon and D V B-S2 c ases. In order to ev aluate the tec hnolog y gap, we defin e the difference of OBO b etween NOFR and BH systems as ∆ OBO = x 1 − x 2 . Fig.8 shows ∆ OBO with respect to ∆ η max , which is defined in ( 43). W e can see that ∆ η max is alm ost linear with ∆ OBO , an d the slope is increasing with BH system OBO ( x 2 ) incr easing. T his result is very useful to p redict the technolog ical gap between NOFR and BH systems. Fig.9 shows the distribution of th rough put f or n - order dif- ference cost fu nction an d fairn ess co st function along K = 50 9 beams that h av e a linear distribution traffic dem and. I n this simulation, we assume that β = 3 × 10 7 , n = 2 (second order function), N t = 32 , N re max = 8 , the SINR γ k and the weighting factor ω k are con stant for all the cells in order to simplify . The result shows that two d ifferent cost func tions distribute the total available resour ce (car riers or time/space) to all the ground cells with different p attern. Fairness cost function is more fa vorable for low traffic requ irement cells while n -order c ost function distribute more resourc e to high traffic requ irement bea m. The perf ormanc e of BH is slightly better than N OFR, especially fo r the low traffic r equiremen t beams. Further, the n - order simply neglect too low-loaded beams. This is relevant result since it is already con sidered in satcom design. V I I I . C O N C L U S I O N S Current design s o f broad band satellite sy stems are lac k of the necessary flexibility to match realistic asymmetric traffic distributions. T wo new techn ologies are stud ied to replace th e current ones over m ultibeam satellite systems. W e prove that the two technolo gies a re dual of each other in freq uency and time domains. Mor eover , the technological gap between NOFR and BH systems is form ulated. T wo novel capa city optimiza- tion prob lems, P1 an d P2, a re in vestigated to best match the individual SINR constrain ts. T he curren t state-of-the art PHY layer technolo gy: D VB-S2 and Shanno n are implemented in order to evaluate the performan ce. The results show significant improvements in terms of power gain, spectral effi ciency and traffic m atching ratio c ompared to the con ventional system. For a DVB -S2 an d K = 20 0 case, we c an ach iev e up to 3 dB power g ain, 0 .7 bit/s/Hz spec tral efficiency g ain, and im prove 10 % traffic m atching ratio by th e proposed cap acity optimizing approa ch. For the du ality stud y , the results show that the technolog ical gap is only related to the OBO of NOFR and BH, and the ga p is almost linear with ∆ OBO . Fur ther , we solve the second pro blem P2 with different c ost f unctions. Fairness cost fun ction is more fav or able for low traffic r equireme nt cells wh ile n - order cost function d istribute more resou rce to high traffic requ irement beams. 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