Statistical Approaches for Initial Access in mmWave 5G Systems

Statistic al Approaches f or I nitial Access in mmW a v e 5G Systems Hossein Soleimani, Ra ´ ul P a rada, Stef an o T omasin and Michele Zorzi Dep. of Information En gineering University of P adova, Italy { soleimani, rpar ada, tomasin, zo rzi } @dei.un ipd.it Abstract —mmW a ve communication sy stems ov ercome high attenuation by using multipl e antennas at both the transmitter and the recei ver to perform beamf orming. Up on entrance of a user equipment (UE) into a cell a scanning procedure must be performe d by the base station in order to fin d the UE, in what is known as i n itial access (IA) proce dure. In t his paper we start fr om the o bserv ation that UEs a re mor e likely to enter from some directions than from oth ers, as they typically move along streets, while other movements are impossible due to the presence o f obstacles. Mor eover , users are entering with a give n time statistics, for example d escrib ed by inter-arr iv al times. In this context we propose scanning strategies for IA that take into account the entrance statist i cs. In particular , we pro p ose two app roaches: a memory-less rand om i llumination ( MLRI) algorithm and a statistic and memo ry-based illu mination (SMBI) algorithm. The MLRI algorithm scans a random sector in each slot, based on the statistics of sector entrance, wi thout memory . The SM B I algorithm instead scans sectors in a deterministic sequence selected according t o the statistics of sector entrance and ti me of entrance, and taking into account the fact th at th e user has not yet been discov ered (thu s includ ing memory). W e assess the perfo rmance of the proposed methods in terms of a vera ge di scov ery time. Index T erms —mmW av e, 5G, Init i al Access, S tatistical, memory-less and memory-based algorithms. I . I N T RO D U C T I O N At the e n d of 201 6, the glob al mo bile d ata traffic was increasing by 7.2 exabytes p er month [ 1] and an incremen t up to 4 9 exabytes mo nthly is expected b y 202 1. Unfo rtunately , traditional microw av e cellular bands will not b e suitable to accommo date th is amou nt of data. Hen ce, there is a ne ed to study alternati ve technologies to deliver the constant incre ment of data y e arly . T h e millim e ter wa ve (a.k.a. mm W a ve) band from 30 to 300 GHz is a p romising space to allocate data. Howe ver, a drawback is th e hig h attenua tio n in non-line- of- sight (NLOS) scen a rios as the millimeter wav es are b locked by common o bstacles such as trees, buildings and h u man bodies. In a heterog eneous deploym ent, comm on macro b ase stations (MBS) fr om typical long-term ev olu tion (L TE) n etworks will be co mplemen ted by sma ll cells to provide the desired Gig a- bits spee d rate. Opposite to L TE MBS which typically transm it in an isotropic manner, mmW ave small b a se stations (SBS) must tra nsmit throu gh directiona l anten nas to increase the signal to noise ratio (SNR), covering a range o f u p to 2 00 This work has been support ed by the Resear ch Labora tory of Huawei Milan , Italia . meters. Similarly , th e user equipment (UE) will have multiple antennas an d will b e able to receive signals comin g from the SBS only if the UE’ s b eamform e r is directed towards the SBS. When a UE enters th e cell, neither the UE nor the SBS know where the other terminal is, therefo re they do not know wh ich beamfor mer must be u sed to co mmunicate . An in itial access (IA) procedur e m u st be started, in which the SBS seq uentially sends pilot p a ckets in various directions and in the me a n time the UE scan s the beam forming directions in o rder to rec eiv e them. Th e proced ure is completed whe n the SBS beamfor mer points to the UE and the UE beamfo rmer points to the SBS. Since the cell discovery time is an issue in mm W a ve networks, several appr oaches have been prop osed to minimize the time wh en a user is found by the SBS. Desai et al. [2] discussed these issues from the beamfor ming procedu re point of v iew . Jeong et al. [3] compared b o th o mnidirectio nal and dir e ctional search techniques. Ferran te et al. [4] stud ied the effects in terms of signal to interferen ce plu s noise ratio ( SINR) du ring the IA when the UE is in rotational motion. They c o mpared the performa n ce o f two alg o rithms, exhaustiv e and staged: the exhaustive algo rithm scans the whole 36 0-degree space in a sequ ential m anner while the staged appro a c h defines thinn er b eams into wider ones. Barati et al. [5] introdu ced the con cept of gen eralized likelihoo d ratio test (GLR T) where base station s pe r iodically transmit synchro n ization signa ls to establish commun ication. Capo n e et al. presented cell discovery schemes based on loc ation informa tio n [6]; in [ 7] th ey also introduce d a le a r ning ap- proach to reduce the d iscovery time b y taking into accoun t the p resence of ob stacles comm only fou n d in real scenarios (i.e., buildings). Qi and Nekovee [8] pro posed a coo rdinated initial scheme for standalo ne mmW a ve networks based on the power delay profile. Gio rdani et al. c o mpared different cell discovery schemes [9] to find the b est trade-off between delay an d coverag e . Th ey studied the perf ormance of state- of-the- a r t IA schemes such as exhaustiv e and iterative (a.k.a. staged). Further more, they included a context-inf ormation based schem e where glob al positionin g system (GPS) coor- dinates f rom the closest SBS are provided by the MBS to the UE. Th e UE also gets its GPS infor mation to steer th e beam to the closest small ba se station. Th is approach reduces the delay b ut in c reases the energy con sumption. Ab b as and Zorzi [10] stud ied the previously stated IA algo rithms in terms of energy consumption for beam f orming considering both a lo w power and a high power analog- to-digital converter . T hey also present in [11] an analo g b eamform ing receiver architectur e based on co ntext information at the user . An artificial neur al network (ANN) appro ach is p resented in [12] to incr e ase the positioning acc u racy . The r e are few works b ased on a geo- location d atabase [13], [14] where , by accumulating the UE position using GPS f or a mo r e ef ficient IA proced ure. Lu o et al. [15] propose a new a nd auxiliary transcei ver for IA which operates in a narrow subban d r educing the transmission leng th of beaco n and as a consequen ce th e beam alignment. Park et al. [16] estimate the user location fr om f ew mm W ave access points (AP) and optimize the beamwidth re quired by selecting the best AP to a given user . L i et al [17] analyze the tw o - stage beamfo rming appr oach ag ainst the exhausti ve sch eme resulting in an incre m ent of th e user-perceived th r oughp ut. W ei et al. [ 18] prop ose a hybrid IA schem e compo sed by a two-stage training wher e the SBS trains in the first stage and a reverse trainin g is p erform e d by the UE in the second stage. In [19] the Neld er Mead metho d is employed for fast and low complexity IA solutions. Parada and Zorzi p ropose a learning approa c h to reduce the delay in cell discovery procedu re [20] by prio ritizing those sectors with higher detectio n p robability using the historical UE’ s location infor mation. L i et al. [ 21] study different IA proto cols, including L TE, to find the optimal design to re d uce the cell discovery d elay w h ile obtaining the h ighest user-perceived downlink throu ghput. Habib et al. [22] propo se a hybrid algo r ithm comp aring it with both the exhaustiv e and th e iterative state-o f-the-ar t technique in terms of misdetectio n prob ability and discovery d elay . In this pap e r we start f rom the observation that UEs are more likely to enter fro m som e directions than other s. Th is is due to th e fact that u sers typically move along streets, while other movements are impossible due to th e presenc e of obstacles like buildings and walls. Note a lso th a t some streets are mo re trafficked th an others, further unbalan cing the probab ility that users enters from specific directio ns. Moreover , users are entering with a given time statistics, for examp le described by an inter-arriv al time between two users. The tim e statistics are also r elev ant for the cho ice of a beamfor ming scanning sequence by the SBS. In this context we adop t IA scanning strategies that take into acco unt the entrance statistics. I n particular, we pro pose two approa ches: a me m ory-less rand om illumination (M LRI) algorithm an d a statistic an d m emory- based illuminatio n (SMBI) algor ith m. The MLRI algorithm scans a random sector in each slot, based on the statistics of sector entrance , withou t memory . The SMBI algorith m instead scans sectors in a de te r ministic sequence selected a ccording to the statistics of sector en trance and time of entra n ce, and takin g into acc o unt the fact that the user has not yet b een discovered (thus including memory ). W e derive the optimal bea mformin g scannin g seq uence fo r bo th approa c h es that minimize the a verage discovery time. Then we assess the perfor m ance of the prop o sed methods in terms of discovery time. The remaind er of this p aper is o rgan ized as follows: Section II provides the system model and describ es the IA prob lem an d the average d iscovery time metric. The proposed algorithms are described in Sectio n III. Simulation experiments an d results are d escribed in Section IV. Fin ally , we con clude the paper in Section V. I I . S Y S T E M M O D E L W e consider a mm W ave cellular system and fo cus on the pr o blem of IA. Both base station (BS) and UEs are equippe d with m ultiple anten nas. T o this en d, we suppo se that the (sign al) space is divided into N sectors th a t can be separately illuminated by th e BS in order to discover new UEs. Secto r illumination is perf ormed by suitably choosin g beamfor mers that are ap plied to the tran smitted sig nal. In particular, the BS will be transmitting a packet kn own to all users, containing also th e in dex of the illuminated sectors, and the users attem pt to decode it cho osing in turn a suitab le beamfor mer to illuminate the d irection from which the signal is comin g. Sectors ar e non-overlappin g and com p letely cover the c e ll. Note that the case of overlapping sector s can be easily accom modated in o ur work, leadin g only to a more complicated mathema tical notation. W e indicate with p i the discrete probability de nsity function ( PDF) of the sector o f entrance of the generic u ser, i.e., p i is the prob ability th at user enters th e cell from sector i = 1 , . . . , N . T ime is divided into slots and the scanning procedu re by the BS is per formed by explo r ing sector b k in slot k u ntil the user is fou nd. W e also assume that within o ne slo t the user explores all the directions choosing multiple be a mformer s, so that when th e BS is illu minating the sector in which th e user is, the user is discovered and the IA pr ocedure fo r th at user is termin a te d . No te that we ign ore the effect of noise, chan n el fading and inter ference that co uld prevent the detection of the IA packet by the u ser , thus d elaying or pr ev e n ting the user discovery . Mor eover , we ignore false alar m events, i.e . , cases in which the u ser erroneo usly detects the IA packet (wh en it has not been transmitted or with the wr ong in dex of the illuminated sector) . All these features can b e achie ved with high p robab ility b y su itably cho osing a mo dulation an d co ding format fo r the IA packet. W e fur th er assume th a t the u ser remains in the sector from wh ich h e entered th e cell at least until it is discovered, i.e., the I A procedur e is faster than th e user movement within the cell. Lastly , we assume that at m o st one new user is present in the system, i.e., we neglect multiple entrance until th e fist u ser ha s been discovered. The time slot of entrance of the user is a random variable, with PDF w k , k = 0 , 1 , . . . , ∞ . W e denote with τ the discovery time, i. e., the number of slots that inter ven e b etween the entrance of the user into the cell and its d iscovery of the BS, i.e., the end of the IA p rocedu re. No te that τ is a random v ariable, depending on the explor ation sequ e nce o f the sectors b y the BS and the sector of entrance o f th e user . In particular, we look f or sector exploration strategies that minimize th e a verag e discovery time E [ τ ] . In particular, let z k,t be the p robability of d iscovering users in slot k , g iven that it entered in slot t , then the average discovery time is ¯ τ = E [ τ ] = ∞ X t =1 ∞ X k =1 ( k − t ) z k,t (1) I I I . I A A L G O R I T H M S In this section, we introdu ce two IA algorithm s: a mem ory- less rand om illumin ation (MLRI) algorith m an d a statistic and memor y-based illum in ation (SMBI) algorithm. The ML RI algorithm scans a ran d om sector at each slot, based on p i and without memory . The SMBI algorithm in stead scans sectors in a d eterministic sequence selected accordin g to the statistics p i and w k , and tak ing into acco unt the fact that user has n ot yet b een discovered (thus includin g m emory) . A. Memory-less Ra ndom Illumina tion A lgorithm The ML RI algorith m random ly chooses the sector to be illuminated at each slot using the PDF q i , i = 1 , . . . , N , to b e prop erly o ptimized. The algorithm does no t have a ny memory of th e illumin ated sectors, as e a ch ra ndom choic e is indepen d ent of the previous ones. In this case, the pro bability of discovering the user in slot k g i ven that it entered in slot t is z k,t = N X i =1 p i (1 − q i ) k − t q i . (2) The PDF q i that minimizes the av erage discovery time is obtained by solving the problem min { q i } E [ τ ] subject to N X i =1 q i = 1 q i ≥ 0 . (3) W e solve the pro blem using the Lagran gian function f ( { q i } , λ ) = k X t =1 w t ∞ X k = t N X i =1 p i (1 − q i ) k − t q i ( k − t )+ + λ N X i =1 q i − 1 ! , (4) where λ is the Lagr a nge multiplier, and with a simple change of variable we hav e f ( { q i } , λ ) = ∞ X k =0 N X i =1 p i (1 − q i ) k q i k + λ N X i =1 q i − 1 ! , (5 ) Computing the deriv ative o f the Lagrang ia n fun c tio n with respect to q i provides ∂ f ( { q i } , λ ) ∂ q i = − p 2 i q 2 i + λ = 0 , (6) and fro m the un itary-sum constraint we immediately h ave q k = p k . (7) Therefo re we rand omly select the sector b k to b e illuminated in slot k acco rding to the PDF of the sector of entran ce p i . For the ML RI algor ithm, the a verage discovery time is therefor e ¯ τ MLRI = N X i =1 p 2 i ∞ X k =1 k (1 − p i ) k = N X i =1 (1 − p i ) (8) B. Statistic and Memory-based Illumina tion Algorithm The MLRI a lg orithm does not have memory o f previously explored sectors, as b k are ind ependen tly g e nerated with a slot-in variant PDF . Instead, the SMBI alg orithm illuminates sector b k in slot k based both on the statistics o f users entrance ( p i and w k ) and o n the fact that the user has not yet been discovered in slot k . In or der to determ ine the optimal sequ ence of sector illumin ation b k that m in imizes the av erage discovery time, we resort to th e m a x imum-a- posteriori criterion, i.e., we maximize the probab ility of discovering the user given tha t we did not fin d it before. T o this end, define the set of explored sectors between slot t and slot k as S ( k , t ) = { i : ∃ t ≤ ℓ ≤ k : b ℓ = i } , (9) i.e., sector i is in S ( k , t ) if ther e exists at least on e slot between t and k in which sector i has been illumin ated. Now , g i ven that the u ser h as not ye t been d iscovered by slot k , and that the pr eviously explored sectors are b 1 , . . . , b k − 1 , the sector to be illuminated in slot k in order to minimize the average discovery time is the on e maximizing the pro bability of finding the u ser in slot k (given p rior assump tions). Th erefore, the probab ility of fin d ing the user in slot k illuminating sector ℓ is v k ( ℓ ) = k X t =1 w t p ( t ) ℓ P N i =1 p ( t ) i , (10) where we set to zero th e prob a bility th at the user is in sector ℓ if we did n ot find it in this sector a f ter it was entered, i.e., p ( t ) i = ( 0 i ∈ S ( k − 1 , t ) p i otherwise (11) The sector to be illum inated is then selected as b k = a rgmax ℓ v k ( ℓ ) , (12) and this ch oice ca n b e perfo rmed sequentially , from the first slot ( f or which b 1 is the secto r having th e maximu m probab ility p i ), to the next slots. The resulting sequence is a deterministic, therefore as lon g as th e statistics p i and w i do not change we will use sequence { b k } obtained from (1 2) to discover a ll users in the system. I V . N U M E R I C A L R E S U LT S W e now assess the p erform ance of the p roposed IA algo - rithms. W e assum e th at we have N = 17 secto rs. For com pari- son purpo ses we a lso consider th e exhausti ve-search algorithm (EA), that illum in ates sectors using a periodic sequ ence b k , where a period is a rando m p ermutation of the sector ind ices 1 , . . . , N . In this case the average discovery time is ¯ τ EA = ∞ X i =0 i   i − 1 Y j =1  N − j − 1 N − j     1 N − i  ≈ N 2 (13) The distrib ution p i depend s on the topolog y o f the cell, th e presence of do minant paths for the users du e for example to the p resence of streets and obstacles like building, together with user habits. This distribution can either obtained from the topo logy o f the cell and side info rmation on user b ehav- ior , or estimated by the BS, based on the IA pro c edure o f previous users. Here for d emonstration p urposes we con sider a equ ilateral triangular PDF p i with pa rameter L , i.e., p u =      4 L 2  u − [ N 2 − L 2 ]  uǫ  max { N 2 − L 2 , 1 } , [ N 2 ]  4 L 2  1 − [ N 2 − L 2 ] − u  uǫ  [ N 2 ] , min { N 2 + L 2 , N }  0 otherwise (14) where users n ev er enter fro m sectors ou tside the interval [ N / 2 − L / 2 , N/ 2 + L/ 2] and the prob ability in this in terval has an equilateral triangular shape. Fig. 1 shows some examp les of the d istribution p i for thr ee values of L . Th is allows to assess the p erforma nce o f th e pr o posed algorithms wh en the PDF o f entrance sector is mor e or less co ncentrated . For L g oing to infinity we obtain the u niform PDF , i.e. , p i = 1 N while for L = 0 we have tha t users only en ter from sector N / 2 . Also the the PDF w k of the user time of entrance, suitable models could be provided or w k can be estimated by observa- tions o f the BS. Here we consider an expon entially distributed inter-arriv al time between user entr ances, therefor e w k = e − µk 1 − e − µ , (15) where µ is the a verage inter-arri val time. The perfo rmance of the various schemes is assessed in te r ms of the average discovery time (1 ), for which we consider both the av e r age a nd the PDF . As alr eady stated, we ig nores the random effects o f th e channel an d th e false alarm s and missed detection pro bability of user discovery . 2 4 6 8 10 12 14 16 i 0 0.05 0.1 0.15 0.2 0.25 0.3 p i L=22 L=12 L=6 Fig. 1. Sector entrance distrib ution p i . 5 10 15 20 25 30 35 L 0 2 4 6 8 10 12 14 16 EA MLRI SMBI Fig. 2. A verage discov ery time as a function of L with µ = 0 . 1 . In Fig. 2 we first assess the av erage d iscovery time ¯ τ = E [ τ ] as a fu nction of th e parameter L of the PDF p i , fo r a fixed value of µ = 0 . 1 . A Monte-Carlo simulation h as been r un to obtain the re su lts for the various algorithm s. W e can observe how the exhaustive scheme (blu e-circle line) remains constant at 8 slots. The ML RI algorithm starts with a discovery time of 0 sinc e for L = 2 there is only one sector with pro bability one where the user could enter, th erefore also q N/ 2 = 1 . A similar observation ho lds f or the SMBI alg orithm that also shows a zero av er age discovery time. Then we o b serve that as L increases the average discovery time of both SMBI an d MLRI increases, as the distribution p i becomes more disp e r sed. Howe ver, while th e SMBI has the best pe r forman ce ac h ieving the minimum average discovery time, the MLRI algorithm requires even more time th an the exhau sti ve a p proach for L > 1 0 , as indeed with th e MLRI approac h it is possible that sectors a re explored mor e than on c e in N co nsecutive slots, thus being inefficient as p i tends to be u n iform. 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 5 6 7 8 9 10 11 12 EA MLRI SMBI Fig. 3. A verage discov ery as a function of µ for L = 14 . Fig.s 3 and 4 show th e average d iscovery time in the different value of µ for a large and small triangle window size of L = 14 and L = 8 , respectively . W e o bserve that the SMBI meth od has the lowest average discovery time, than ks to the exploitation of memory . As alread y observed , fo r small L ML RI ou tp erform s EA wh ile for h igh L EA o u tperfor ms MLRI. Moreover both M LRI an d E A show a con stant average with respect to µ while fo r SMBI the average discovery time decreases with µ . 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 3 4 5 6 7 8 EA MLRI SMBI Fig. 4. A verage discove ry as a functi on of µ for L = 8 . Lastly , Fig.s 5-7 sho w the PDF of th e discovery time for the various ap proach es, for µ = 0 . 1 and L = 10 . As we expect, the d ensity f unction f or EA is unifo rm over the set [0 , N − 1] as within N slots all sectors are explored. Both MLRI and SMBI methods instead exhibit a mo re co ncentrated PDF , in particular with the SMBI method being able to discover users within 1 0 slots w ith p robability very close to 1. Th e MLRI method shows instead a more dispersed PDF , as expected f r om the results on the a verage discovery time. 0 2 4 6 8 10 12 14 16 0 0.05 0.1 0.15 PDF Fig. 5. PDF of disco very time for EA algo rithm with L = 10 , µ = 0 . 1 . 0 5 10 15 20 25 30 0 0.05 0.1 0.15 PDF Fig. 6. PDF of discov ery time for ML RI alg orithm with L = 10 , µ = 0 . 1 . 0 1 2 3 4 5 6 7 8 9 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 PDF Fig. 7. PDF of disco very time for SMBI algor ithm with L = 10 , µ = 0 . 1 . V . C O N C L U S I O N S This p a per intro duced two novel alg o rithms to r educe the av erage discovery time for th e IA pr o cedure fo r mmW a ve massi ve MIMO systems. W e compared our propo sed sch emes with the E A and show that SMBI always outperform s EA. 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