Active IoT User Detection in Near-Field with Location Information

JOURNAL OF L A T E X CL ASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 1 Acti v e IoT User De tection in Near -Field with Location Informati on Gabriel Martins de Jesus, Richard Demo Souza, Onel Luis Alcaraz L ´ opez Abstract —In this paper , we addre ss activ e users detection (A UD) in near -field Inter net of Things (IoT) n etworks by explor - ing prior knowledge of users’ locations. W e consid er a scenario where u sers are distributed in a semi-circular area wit h in the Rayleigh d istance of a multi-anten na base station (BS). W e pro- pose th e BS to use location estimates of the u sers to r econstruct their line-of-sight (LoS) channel components, hen ce assisting the A UD process. F or this, th e BS combines these r econstructed channels with users’ pilot sequences, enhancing the corr elation between receive d signals and active users. W e f ormulate the location-aided A UD as a con vex optimization problem, solved via t h e alternating direction method of multipliers (ADMM). Our proposal has a higher computational complexity com pared to the baseline ADMM app roac h where location info rmation is not used. Moreov er , the proposal requires location information of users, which can be readily informed if users are static, or inferred via established localization algorithms if they are mobile. Simulation results compare our proposal against the b aseline acros s varying systems parameters, such as number of users, pilot length and LoS component str ength. W e d emonstrate that under perfect location estimation and strong LoS, our p roposed method significantly outperfo rms the baseline. Furthermore, r obustness analysis shows that performance gains persist under imperfect location estimation, p ro vided the estimation error remains wi thin bounds determined by the system parameters. Index T erms —In ternet of Thi n gs, near -fi eld communications, activ e users detection, alternating direction method of multip liers I . I N T R O D U C T I O N E LECTR ONIC devices perfo r ming machine type com- munication s (MTC) are beco m ing pr edominan t within modern wireless n etworks [1]. In fu ture 6G networks, MT C will likely be d ominated b y non -critical I nternet of Thin gs (IoT) applications, characterized b y devices with limited com- putational capabilities and strict lo w-power requirem ents [2]. T o accomm odate these con straints, grant-f ree random access schemes are e mployed to min im ize signaling overhead be- tween the base station (BS) and u sers, significantly reducing power consumptio n . While this approach increases the p roba- bility of packet collisions, p o tentially leading to transmission failures, it is specially well suited for the sparse traffic nature of Io T d evices. Since IoT devices are typically configured to G. M. de Jesus and O. L . A. L ´ opez are with the Centre for Wire less Communicat ions (CWC), Uni ve rsity of Oulu, 90014 Oulu, Finland. (e-mail: gabriel .martinsdeje sus@oulu.fi, onel.al carazl opez@oulu.fi). R. D. Souza is with the Electric al and Electronics Engineeri ng Department, Federal Uni ve rsity of Santa Cata rina (UFSC), Florian ´ opolis 88040-900, Brazil (e-mail: ric hard.demo@ufsc.br). This research has been supported by the Research Council of Finland (former Aca demy of Finl and) Grant 346208 (6G Flagship Programme) and Grant 362782 (ECO-LITE), CNPq (403124/202 3-9, 305021/2021 -4) and RNP/MCTI Br asil 6G project (01245.020548/ 2021-07). transmit only wh en new data is acquired , th ey exhibit spor adic activity and are rarely active simultaneou sly [3 ]. This in herent sparsity in MTC network s is a cr itical p roperty exploited during the active u sers detection (A UD) phase. In the A UD phase, the BS aims to detect and iden tif y the specific subset of acti ve users from the total device po ol [4]. Users are assigned p ilot sequences which are attached a s preambles to the ir in formation packets, and these are utilized to identify active users and estimate their respective channe ls based on the received signal. When multiple users transmit simultaneou sly , th e ir pilots arrive at th e receiver sup e r imposed and distorted by th e wireless ch annel, mak ing direc t u ser separation challeng ing for the BS. Howe ver , due to the hig h sparsity of MTC traffic, established detection methods can be used fo r A UD , inclu ding comp ressed sensing ( CS) [5] , ma- chine learnin g [6], and co n vex op timization-based algorithm s such as alternating directio n m ethod of m ultipliers (ADMM) [7]. Accurate A UD is a fund a m ental pr e requisite fo r reliable channel estimation and su bsequent data detection [ 8]. Lookin g beyond 5G, the transition to near-field communi- cations is a p romising scenar io for 6 G tech nologies [9]. As infrastructu re evolves toward increasin g ly large anten na arrays and high er frequen cy band s, th e near-field region extends significantly , ranging from a f ew centim e te r s to hund reds of meters [10] . W ithin th is region , the classical plan ar wave assumptions of far-field m odels are in valid, and spherical wa vefronts are co nsidered instead [11]. Unlike plan ar mo dels, the sph erical wa ve mod el c a ptures phase variations acro ss the array that d epend on both the ang le of a r riv al an d the sp e c ific distance of the source. Due to th ese propag ation character is- tics, tech niques such as beam -focusing in the d ownlink [11] and user localization based solely on up link signals [1 2]– [14] are possible. The latter is crucial for this work, an d this localization capability can be explo red to enhance the A UD prob le m , as by determin ing a user’ s p osition, the system can reco n struct the d eterministic line of sight (L o S) chan- nel response, which typically serves as the dom in ant signal compon ent. This LoS co m ponent, in tu rn, can b e co mbined with the pilot sequ e nce to better differentiate between users, improving the accuracy of the A UD. A. R elated work A UD is typically paired with channel estimation, an d solved jointly . T o achieve better p erforma nce in d e tection accuracy , many works pro posed mo d ifications to established CS alg orithms [1 5 ]–[19 ], such as th e or th ogonal matching pursuit (OMP) [20] an d app roximate message passing (AMP) JOURNAL OF L A T E X CL ASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 2 [21]. In each of th e se works, the propo sed modification s take into acco u nt specific system mod el assumption s, such as de te c tion of the same user in different frequ ency channels [15], spatially correlated ch a nnels [16] , users clustered b y non-o rthogon al pilo t sequ ences [17], as well as including protection fr om m a licious attacks via pilot contamin ation [ 18] and specifically tailor them for satellite networks [19]. A UD has also been solved via Bayesian learn ing m ethods [22] and neural networks [2 3]–[2 6]. Interesting ly , altho ugh th e typical formu latio n of A UD is a non -conv ex proble m , it can b e relaxed to be solved with co n vex optimization algo rithms [ 7], [27]– [29]. Meanwhile, if th e ch annels have been pr eviously estimated, this inf ormation can be used to imp rove the A UD and assist th e data detection. For instance, in a multi-car r ier setup, Y ang et al [30] do n ot consider transmission of p ilot sequences, but instead rely on the ch annel state informatio n (CSI) an d kn owledge of th e sp read sequen ces to estimate which users are ac tive. T h e initial detection is made based on an en ergy test of the received signal in each carrier , declaring activ e the users that transmit in th e car riers that are not idle. Then, a false-alarm me chanism is prop osed, iteratively removing users b ased on the symbols detecte d by the AMP- based data detector, significantly d ecreasing false-alarm ra tio and symbo l error rate. In the far -field, the A UD pro blem h as been extensi vely ex- plored, while its study in the nea r-field has increased recently [31]– [35]. W ang e t al [31] model the near -field ch annel with correlated Rician fading. Th ey form ulate A UD as a m aximum likelihood estimation (M L E) problem , which is sign ificantly more complicated than with unco rrelated channe ls. T h e per- forman ce of the pr o posal is evaluated ag ainst the ty p ical uncorr elated chan nels assumption s, outperf orming th e latter as the total numb er of users increase or the pilot leng th decrea ses. Mylonop oulos et al [3 2] propo sed a techn iq ue that jo intly detects the acti ve u sers and estimates their positions with the aid of several recon fig urable intelligent su rfaces (RIS). Here, users are not assign ed individual p ilot sequ e n ces a pr iori, risking collisions. Each RIS is d esigned to inspect a specific area, attempting to identify the spatial origin of the incoming signals. Then , reso u rce blocks are a ssign ed to the u sers in those area s, and multip ath compon ents are suppressed by the RIS and do not reach th e BS. Comp ared to the case without RIS, the pro posal imp roves per f ormance w h en there are non- LoS (NLoS) compo nents as the symbols’ power increases. Arai et a l [33] split A U D in two ph ases based on OMP, first estimating the chann els, then detecting the users an d associat- ing the chann els. In the for mer , an OMP algorithm is ran over a polar grid covering the deployment area of the u sers. Then, the detected channels are associated with the users. The propo sal guaran tee s lower n ormalized m ean-square d error and bit error rate than oth er appro a c hes. Qia o et al [34] use an extension of OMP to p erform A UD and channel estimation, and also propo se an algorithm to estimate the position of the ac ti ve users. The chan n el estimate of the active users is used with the MUSIC algor ithm to estimate th e angle of arr i val an d time difference of arriv al and obtain the coo r dinates of the users with the least squares algor ithm. A UD and channe l estimate perfor mance is superior to other baselines, and the po sition error decreases with higher transmit powers. Zhan g et al [35] explore the spatial stationary of the near-field by mod eling the activity m atrix as a three-level sparse matrix, in volving the acti ve state, location o f users, and visibility regions. They propo se two algorithms th a t do channe l estimation and A UD, either jointly o r separately , o utperfo rming the baselines with the cost of inc reased compu tational co mplexity . While prior work s in the far-field h av e utilized ch a nnel knowledge to improve A UD performance [30 ] , to the best of our kn owledge, this appro ach has not yet b een explored in the near-field context. T o add r ess this gap , our paper makes th e following main contributions: 1) W e f ormulate A UD as a c o n vex optimization problem that explicitly inc orporates p rior k nowledge of users locations. While [ 13], [14] have demonstrated that u ser location can be o b tained in th e near-field with reasonable accuracy , and [32]–[35 ] obtain estimate s of the location of users as a consequ ence of A UD, n one use location informa tio n as pr ior in f ormation to imp rove A UD accu- racy . 2) W e solve the A UD with location information via ADMM. Usin g ADMM to solve the A UD h as b een explored in the literatur e [ 7], [16] , [36 ], but explicitly using th e chan nel matrix adds more co mplexity to the formu latio n, and we deriv e the closed fo r m expression s for obtain ing th e set of activ e users. 3) W e sh ow that the propo sal retain s its superior ity even with imperfect estimates, provided the estimation error remains below a specific network-dep endent thre shold. Moreover , our propo sal o utperfo rms the baseline in sce- narios with a larger nu mber of total user s, active users, and an tenna elem ents, wher eas the baseline is fav ored by longer pilot sequ ences when lo cation in formation error s are high . Furthermo re, o u r results ind icate that while the baseline is pref erable in pu re NLoS channels, our propo sal achieves superior perform ance provided th at the LoS co mponen t between the users and th e BS is sufficiently stron g. The remaining o f this paper is organized as follows. In Section II, we in troduce our system model, detailing all as- sumptions considered in this work. In Section II I, we introd uce the co n vex optimizatio n p r oblem an d present the solutions via ADMM for the c ases with kn own and unk nown users location . In Section IV, we ev aluate the performa nce of ADM M with perfect location in f ormation (ADMM- LI) ag ainst the baseline without loca tio n infor m ation under different ne twork and esti- mation para meters. Lastly , we con clude the pap er in Section V. Throu g hout this pap er , we a dopt the following notation . Matrices are denoted by A , and vectors by a . For m atrix B , its i -th row (or column) is deno ted by b i , and the elemen t at the i -th row an d j -th colum n is denoted by B i,j . For matrix A , || A || F denotes its Frobeniu s. The diag ( c ) oper ator takes vector c and tra n sforms it in to the m atrix C , with eac h diago nal element given by the elem ents in c , and o ff-diagonal eleme n ts equal to 0, while the vec ( D ) o p erator takes the P × Q matrix D and flattens it into the 1 × P Q vector d = [ d 1 , d 2 , . . . , d P ] . JOURNAL OF L A T E X CL ASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 3 I I . S Y S T E M M O D E L W e con sider an IoT network with N single-an tenna devices served b y a sing le BS that is centered at the orig in, as illustrated in Fig. 1. T he d istance of the n - th user to the or igin is d enoted by r n , and its angle to the X-axis is deno ted b y θ n . The users are p lac ed in a semi-circu lar area, and are within the region limited by r min ≤ r n ≤ r max , and θ min ≤ θ n ≤ θ max . In cartesian coordin ates, p n = [ r n cos θ n , r n sin θ n ] denotes the position of the n -th u ser . The BS has an e stima te o f the position o f u sers, but error s in th e estimate a r e pr esent in practical deploym ents, which we m o del here by intr oducing the random variables r err ∼ N (0 , σ ) and θ err ∼ U (0 , 2 π ) that are co m bined and summed to the true p osition of the users. The r esulting is a p osition estimate ˆ p n that is in a p oint in the circumfer ence o f radius r err and center p n , i.e., ˆ p n = p n + [ r err cos θ err , r err sin θ err ] . (1) At each time slot ( TS), any user may gen erate a packet with probab ility p = K/ N , and transmit it in the shared frequen cy channel. W e d e note by K th e set of active users. The BS is equippe d with an a n tenna array of M elements center ed at the origin and with each element m lo cated in δ m . Each elemen t is spaced fro m the oth ers b y d istance of at least d = λ/ 2 , where λ = c/f is the w av elength of the carrier, and c is the speed of light. A. Ch annel model W e consider that the user s are in the radia tive field an d their distance is below the Ray leigh distance d R , 2 D 2 /λ , i.e., 0 . 62 p D 3 /λ ≤ r n ≤ d R [11], w h ere D denotes the dimension of the an tenna array and dep ends on its topolo gy , as well as n umber of antenn a elements. In such region , users experience near-field effects. In the near-field, if the antenna elements are sufficiently close, the mag nitude of the received power of a given signal can be mod eled as eq ual in all M elements. Howe ver , the phase is nonlinear and depend ent on the distance from the user to each elemen t [11 ]. Let h LoS n,m denote the LoS channe l between the n -th user and the m -th antenna element, such th at h LoS n,m = α n e j ϕ n,m , (2) with α n = λ 4 π r n , (3) and ϕ n,m = − 2 π λ r n,m , (4) where th e term r n,m = || p n − δ m || 2 is the distance between the n -th u ser and the m - th antenn a element in the ar ray . The channel respon se is arran ged in a chann el vector as h LoS n = { h LoS n,m } M m =1 , and th e ch annel matrix is d efined as H LoS = [ h LoS 1 , h LoS 2 , . . . , h LoS N ] ⊺ ∈ C N × M . (5) Moreover , the NLoS compo nents follow a complex Gaussian normal d istribution, with each elem ent de n oted by h NLoS n,m , and arrange d in the chann el vector h NLoS n . Th e vectors, in turn , are arrange d as H NLoS = [ h NLoS 1 , h NLoS 2 , . . . , h NLoS N ] ⊺ ∈ C N × M . Fig. 1: Illustration of the system model, wh e re users a re indicated by the filled circles, with black filling ind icating the K activ e users. W e con sider a Ricean ch annel with p arameter µ , which relates the to tal power of the Lo S to the NLoS compo nents. The resulting channel is g i ven by H total = r µ 1 + µ H LoS + r 1 1 + µ H NLoS . (6) B. S ignal model Before tr a n smitting data, an active user will transmit its pilo t sequence to th e BS. The BS is aware of all sequences, w h ich are unique to each user , and are used to identify the active users and poten tially estimate th eir chan nels. W e deno ted by T the total nu mber o f sy m bols allocated to transmitting the pilot sequ ence, an d we assume th at T is smaller than the coh e rence interval in the time fra m e. Let φ φ φ n ∈ C T × 1 denote th e p ilot sequ e n ce assigned to the n -th user, with || φ φ φ n || 2 = 1 . Then, the pilot sequen ce matr ix is d efined as Φ , [ φ φ φ 1 , φ φ φ 2 , . . . , φ φ φ N ] ∈ C T × N . Moreover, we consider T ≤ N , such that, althou gh uniq ue, the users’ pilot sequen ces are not pair -wise orthogon a l, wh ich results in soft collisions. Unlike in ha r d collision s, where more than a sing le u ser transmits the same pilot sequen ce, it is still p ossible to separate the incoming signals when soft collisions occu r . W e denote by X the acti vity matrix, defined as diag ([ x 1 , x 2 , . . . , x N ]) , with x n = √ γ n if n ∈ K , and x n = 0 otherwise. Th e u sers use in verse power contro l b ased on their location informatio n , such that, γ n = (4 π || ˆ p n || 2 /λ ) 2 is the transmit power of u ser n . This leads to the power o f the received signa l to be unitary if there was only LoS and if ˆ p n = p n 1 . The received signa l Y ∈ C T × M is then g iv en by Y = ΦXH total + V , (7) 1 Although other po wer control approache s (or no powe r cont rol at all) coul d be considered without affe cting the dev elopment of the proposal, we set for this specifica lly to avoi d the risk of the algorithm being biased towa rds users closer to the BS. JOURNAL OF L A T E X CL ASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 4 where V ∈ C T × M is the additive wh ite Gaussian n oise. Another way to write the received sign al is by defining J , ( XH total ) ∈ C N × M , which is a convenient represen tation when doing join t activ e u ser detection and channe l estimation. In this case, the r eceiv ed signal is written as Y = ΦJ + V . (8) I I I . AU D W I T H A N D W I T H O U T L O C AT I O N I N F O R M A T I O N During the A UD phase, the BS attempts to iden tify the subset K of active users in th e cur rent TS, utilizing p rior knowledge of the pilot sequences of the user s pool. In g rant- free access, A UD is fund amental to iden tif y which u sers are transmitting, sin c e the typical handsh ake between BS and users does not h appen. For this r eason, improving th e acc uracy of A UD is imp ortant, a s the in formation packets from un de- tected u ser s are lost, a n d noise can be interp reted to be real data f rom inactive users. If th e LoS c o mponen t is dom inant in the rec ei ved signal, it can be explore d to improve accur acy in the A UD process, as the re ceiv ed signa l will be correlated with both pilo t seq uence and the LoS chann el of any user a cti ve. If the BS kn ows th e location of the users, either by estimating it, e.g., with MUSIC, o r b ecause users are fixed and their locatio n is informed to the network, it can r econstruct an estimate of the LoS compon ent. The n , th ese estimated co mponen ts can be combined with the pilot sequences to b etter d ifferentiate between user s. I n this section , we p r opose improving A UD by explicitly incor porating location in formation , formulated as a conv ex p roblem and solved using ADMM. For co mparison, we also present the standard ADMM-based A U D form ulation without loc a tio n data, as d etailed in [7 ]. Thro ughou t th e pap er , we refer to our prop osal as ADMM-LI, and to the form ulation without location info rmation as the baseline. A. P r o blem formulation T o f ormulate our pro b lem, we assume the BS is aware of the loca tio n of users and c o nstructs the matrix ˆ H with each entry accord ing to (2), but with α n = 1 . By igno ring the NLo S compon ents, we model the A UD p roblem as minimize X 1 2 || ΦX ˆ H − Y || 2 F + β || X || 2 , 0 , (9) where β is a regular ization p arameter to eithe r emp h asize the sparsity of th e estimated X or the acc u racy of ΦX ˆ H − Y , and || X || 2 , 0 , P N n =1 i ( || x n || 2 ) , with i ( a ) b e in g the indicator function with value 0 if a = 0 and 1 otherwise. The problem in (9) is no t c on vex becau se o f the term || X || 2 , 0 . As in [7 ], the objective function nee ds to b e relaxed. While we can relax the ℓ -0 norm to ℓ -1, it will be biased to- wards entries in X with high m agnitude. An alter n ativ e is the log-sum pen alty , resulting in P N n =1 log ( u n + ǫ 0 ) , where || x n || 2 ≤ u n , n = { 1 , 2 , . . . , N } , and x n and u n denote the n -th row a nd entr y of X and u , resp ecti vely . Then, we consider the majority -minimization app roximation , obtaining P N n =1 ν ( r ) n || x n || 2 , with ν ( r ) n = ( ǫ 0 + || x n || 2 ) − 1 . The relaxed problem is written as minimize X 1 2 || ΦX ˆ H − Y || 2 F + β N X n =1 ν ( r ) n || x n || 2 , (10) and ν ( r ) n is updated after solving X each time, un til the R -th iteration. B. S olution via ADMM The ADMM is an algor ithm to solve con vex optim iz a tion problem s, inc luding A UD [7] , [16] , [36 ] , th at combine s the separability o f du al ascent methods and the co n vergence proper ties o f the method of multiplier s [37] . ADMM requires that the o bjectiv e function is separable, which is the case in (10). T o app ly it, we split X into X and Z , an d write the equiv alent problem as minimize X , Z 1 2 || ΦZ ˆ H − Y || 2 F + β N X n =1 ν n || x n || 2 sub ject to X = Z . (11) W e defin e the Lagr angian dual variable W ∈ C N × N and obtain the aug mented Lagrangian of (1 1) as L ( X , Z ; W ) = 1 2 || ΦZ ˆ H − Y || 2 F + β N X n =1 ν ( r ) n || x n || 2 + ρ 2 || X − Z + W /ρ || 2 F + || W || 2 F 2 ρ , (12) where ρ > 0 is a penalty parameter . Then, the prob lem is solved by iteratively minimizing the gradient of L ( X , Z ; W ) for Z , then X , and lastly upd ating the d ual variable W with ρ as the step size. At each iteration s , until s = S , th e v ariables are solved and upd ated as vec( Z ( s +1) ) =( ˆ H ˆ H H ⊗ Φ H Φ + ρ I N 2 ) − 1 vec( ˆ H H YΦ H + ρ X ( s ) + W ( s ) ) , (13) X ( s +1) n,n = d n | d n | max 0 , | d n | − β ν ( r ) n ρ ! , (14) for n ∈ { 1 , 2 , . . . , N } , with d n = Z ( s +1) n,n − W ( s ) n,n /ρ and W ( s +1) = W ( s ) + ρ ( X ( s +1) − Z ( s +1) ) , (15) as it is detailed in the Appen dix. C. Case without location information When n o lo cation info rmation is present, the A UD is solved by form ulating the problem as [7] minimize J 1 2 || Φ J − Y || 2 F + β || J || 2 , 0 , also relaxing and using the majority-m inimization a p proxim a- tion, obtain ing minimize J 1 2 || ΦJ − Y || 2 F + β N X n =1 ν ( r ) n || j n || 2 , (16) with ν ( r ) n = ( ǫ 0 + || j n || 2 ) − 1 . T o apply ADMM, the prob le m is rewritten as minimize J , Z 1 2 || ΦZ − Y || 2 F + β N X n =1 ν ( r ) n || j n || 2 sub ject to J = Z , (17) JOURNAL OF L A T E X CL ASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 5 Algorithm 1 ADMM f or A UD Input: Φ , Y , H (ADMM-LI), S , R Output: X / J (ADMM-LI / baseline) 1: for r = 1 : R do 2: ν ( r ) n ← ( ǫ 0 + || x n || 2 ) − 1 / ( ǫ 0 + || j n || 2 ) − 1 3: for s = 1 : S max do 4: Z ( r,s +1) ← (13) / (1 9) 5: X ( r,s +1) / J ( r,s +1) ← (14) / (2 0) 6: W ( r,s +1) ← (15) / (2 1) 7: end for 8: end f or 9: X / J ← X ( R max ,S max ) / J ( R max ,S max ) with the Lag rangian gi ven by L ( J , Z ; W ) = 1 2 || ΦZ − Y || 2 F + β N X n =1 ν ( r ) n || j n || 2 + ρ 2 || J − Z + W /ρ || 2 F + || W || 2 F 2 ρ , (18) and the upd ate rules for J , Z and W a re giv en by [7 ] Z ( s +1) = ( Φ H Φ + ρ I N ) − 1 ( Φ H Y + ρ J ( s ) + W ( s ) ) , (19) j ( s +1) n = d n || d n || 2 max 0 , || d n || 2 − β ν ( r ) n ρ ! , (20) for n ∈ { 1 , 2 , . . . , N } , with d n = z ( s +1) n − w ( s ) n /ρ , and W ( s +1) = W ( s ) + ρ ( J ( s +1) − Z ( s +1) ) . (21) In Algorith m 1, we su mmarize the steps f o r each appr o ach. D. Comp utationa l co mplexity an d implementation In b oth ADMM-L I and the baseline , the operation s with the h igher cost are the matrix in version a n d m a tr ix multiplica- tions for determining Z ( s +1) , ( 13) and (1 9), respectively . For ADMM-LI, the matr ix inv ersion has c omplexity O ( N 6 ) , wh ile the inne r multiplica tio n has complexity O ( N M T + N 2 T ) , and th e resu lting multiplication of the inverted m atrix by the vectorization is O ( N 4 ) , resulting in O ( N 6 + N 2 T + N M T ) = O ( N 6 + N M T ) . On the oth er hand , for th e baselin e , the com- plexity is O ( N 3 + N 2 + N M T ) = O ( N 3 + N M T ) . Althou gh the c omplexity is h ig h, some proper ties o f the matrices in question can be explored to re d uce co mputation al time. First, we note that ˆ H ˆ H H and Φ H Φ are herm itian po siti ve d e finite matrices, an d so is their Kronecker prod u ct and its sum with the identity m atrix. T hus the linear p roblems explicitly solved in (13) and (19) can be mor e efficiently solved with numer ical methods. In practice, the use of LU deco mposition and forward and back substitutio n can sig n ificantly decrea se the run-time of th e algor ithms, althoug h their asy m ptotic comp lexity will remain the same. I V . R E S U LT S In this section, we ev aluate th e perfor mance o f ou r p r oposal and compar e it to the baseline und er several d ifferent n etwork configur ations. I n T able I, we summ arize th e b ase par a m eters, Parame ter Symbol V alue Number of users N 24 Number of acti v e users K 4 Number of antenna s M 32 Carrier frequenc y f 1710 MHz Min. and max. distances of users to BS r min , r max 20 m, 80 m Min. and max. angles of users to X-axis θ min , θ max − 3 π / 7 , 3 π / 7 Pilot length T 6 s ymbols T ABLE I: Summa r y of the base parameters used thro ughou t the simulations. T ABLE II: ADMM hyper-parame te r s fo r the m ethods with known a nd unkn own u sers’ position. Parame ter v alue β 10 − 5 ρ 10 − 1 ǫ 0 10 − 1 R 10 S 10 which are the sam e in all simulation s unless stated otherwise. As our b ase co n figuration , we consider N = 2 4 users and, at ea ch TS, users are active with proba bility p = 4 / N . The value fo r N is c hosen due to th e com plexity of the algor ithms under study , but it can b e representative o f a network with more user s gr ouped into clusters, opera ting orthogo nally in the fr equency , time and/or pilo t sequence do mains. The BS is equippe d with a linear antenna positioned alo ng the Y -axis, with M = 32 e lements, each located at δ m = [0 , ( m − 1) λ − ( M − 1) λ/ 2] , and operates at car rier frequ e n cy f = 17 1 0 MHz, as used, in, e.g., NB-IoT [3 8]. The min imum and maximu m distance of users to the BS are r min = 20 m and r max = 80 m , respectively . Here, the d istances are c hosen such that u sers stay in the radiativ e nea r-field f o r M between M = 32 and M = 64 , as this ran ge will be consider ed in one of ou r exper im ents. Lastly , the minimu m and max im um an gles of users to the X- axis are set to θ min = − π / 2 and θ max = π / 2 , occu pying the first a n d four th quadr ants and facing th e antenna array . W e let the pilot transmission take T = 6 sym bols, and each entry o f the pilo t matrix is drawn fro m a com plex nor mal distribution, with the who le pilot normalized . W e ev aluate th e appro aches using th e co mplement of the balanced accuracy as the ma in pe r forman c e metric, defined as 1 − A = 1 − TPR + TNR 2 . (22) Here, TPR , | T P | /K rep resents the true positiv e rate, wh ere | TP | is the n umber of c orrectly d e tected active users. Con- versely , TNR , | TN | / ( N − K ) is the true negativ e rate, wh ere | TN | de n otes the num ber of c o rrectly iden tified inactive u sers. The h y per-parameters of the ADMM algorith ms are selected a priori such that the resulting perfo rmance is clo se to the best achiev able perf o rmance b ased on training f rom 10 4 iterations, varying for each method . The param eters for each app roach are presented in T able I I . T o achieve reliable and unbiased results, each experim e nt run s f or 10 6 iterations, an d the po sition of th e users, as well as their pilot sequences, are r andomly sampled at each Mon te Carlo iteration. JOURNAL OF L A T E X CL ASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 6 -10 -5 0 5 10 -4 10 -3 10 -2 10 -1 10 0 Fig. 2: Performan ce com p arison between the baseline an d the approa c h with location informatio n as a function of the SNR. A. I mpact of the sign al-to-n oise-ratio ( SNR) W e evaluate th e per forman c e of ADMM-LI as the SNR varies from − 10 to 5 dB and present the results in Fig. 2 for M = 16 and M = 32 ante n nas. While the perfo rmance of both the b aseline and ADMM-LI increases with th e SNR, the p erforman ce g ap is much m ore evident. Already at SNR = 0 dB, ADMM-LI outperfo rms the ba selin e by mo r e than one order of magnitude with M = 16 (dashed lines), and almost two orders of magn itude with M = 32 (contin u ous lin e s) . At SNR = 5 dB, the gap is of mor e than one and two orders of magnitud e for M = 16 and M = 32 , respectively . Ho wev er, at lower SNR values, the sign al is so co rrupted by no ise that the re cei ved sign al do es n ot resemble th e comb ination o f pilot sequence and channel respo nse as much, resulting in mo dest perfor mance im provements. B. Mu ltipath T o ev aluate the impac t of multipath in the perfo rmance of our p roposal, we v ary the Rician fading par ameter from µ = 0 . 5 , mean ing a very weak L oS, to a domina n t LoS c o mponen t with µ ≈ 8 , and present the results in Fig. 3, wher e the cu rves indicated with µ → ∞ stand for the case with o n ly LoS. In the cases with weak LoS, the locatio n of the users provides little information on th e resulting channel, but it is en ough to improve th e per formanc e over the baseline. This is tru e fo r the cases with small or inexistent erro rs, the dashed or ange ( σ = 0 ) and d a shed cyan ( σ = 0 . 5 ) curves in Fig. 3 , which alread y at µ ≈ 1 . 5 provide a perform ance g ain of almo st o ne order of magnitud e c o mpared to the baseline (blue continu o us curve). As the L o S becomes stro nger, th e perf ormance imp rovement is m ore significant, and the curves whe r e µ is varied ap proach their full LoS coun terparts. I n terestingly , as the errors in the location estimate inc r ease, the multipath has less im pact on the per formanc e , with the curves conv erging faster . 1 2 3 4 5 6 7 8 10 -3 10 -2 10 -1 10 0 Fig. 3: Performan ce com p arison between the baseline an d the approa c h es with p erfect and imper fect location info rmation for the c a se with SNR = 0 dB as a f unction of the fading parameter µ . C. Imp erfect location estimate While th e pro posed metho d greatly o u tperfor m s the base- line, assumin g perfect location informatio n is too op timistic in most cases. Th is is possible if u sers are com p letely static and if their positions are precisely and explicitly inf ormed to the BS, wh ic h is n ot always the case. Howev er , it is possible to estimate th e positions of u sers b ased on the receiv ed signal using, e.g. , the M USI C algo r ithm [13], [14 ] or Qiao ’ s [3 4] et al propo sal, a lth ough with so m e errors. T o model these possible errors, we vary σ fr om 0 m (perfec t loc a tion estimation ) to 10 m and presen t the result in Fig . 4 for SNR = 0 d B. The resu lts in Fig. 4 show an inter esting effect: while the perfor mance of ADMM-LI d ecreases as erro rs are introdu ced in the location estimate, this m e thod can still be u sed as long as th e er ror is no t too high , pr ovid ed that th e LoS compon ent is stron g eno ugh. This is evident f rom th e pu rple and g r een cur ves, which shows the cases with strong LoS compon ent ( µ = 5 and µ → ∞ , respectiv ely), outp erformin g the baseline at up to σ = 4 . On the other hand, if the lo c ation informa tio n has high erro rs a n d is not reliab le, it is b etter to use the baseline instead, a s it has better p erforman ce in such cases, and has much lower complexity . Howev er , small errors of around 1.5 m , even f or SNR = 0 dB are reasonable to assume, as Hag hshenas et al [1 3] o bserverd when they ev aluated MUSIC’ s p erform a n ce in a setup sim ilar to ou rs. On top of this, tracking algorith ms [3 9] can furthe r improve the estimates. D. V aryin g pilot length W e ev aluate the perfo r mance o f ADMM-LI as th e pilot length varies f rom T = 2 to T = 1 2 symbols an d present the results in Fig. 5. Alongside the b aseline an d the case with p er- fect locatio n informatio n, we present results fo r σ = { 0 . 5 , 6 } and µ = { 0 . 5 , 5 } . When the loca tion errors are large (cyan curve, σ = 6 ), th e pro posal slightly outperfor m s the ba seline JOURNAL OF L A T E X CL ASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 7 1 2 3 4 5 6 7 8 9 10 10 -3 10 -2 10 -1 10 0 Fig. 4: Performan ce com p arison between the baseline an d the approa c h es with perfect and imperf e ct location inform ation for the case with SNR = 0 dB a s a function of the position error variance. 2 3 4 5 6 7 8 9 10 11 12 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 Fig. 5: Performan ce com p arison between the baseline an d the approa c h es with perfect and imperf e ct location inform ation for the case with SNR = 0 dB as a function of the pilo t length. only in shorter p ilot lengths, with their p erforman ces matchin g at T = 4 , and the baseline o utperfor ming it f o r T > 4 . As the pilo t length in creases, errors in the lo cation estimate are detrimental to the perform ance, as the introd uced err o r in the channel matr ix results in low correlation with the rec e i ved signal. In this case, even if there is o nly LoS, considerin g only the pilot length is mo re ben e ficial to A U D. If location e r rors are small, even with a weak LoS the prop osal outperf orms the baseline, as indicated by the y ellow ( µ = 0 . 5 , σ = 0 ), purple ( µ = 5 , σ = 0 ), gr een ( µ → ∞ , σ = 0 . 5 ) an d dark r ed cu rves ( µ = 0 . 5 , σ = 0 . 5 ) . Moreover , e ven small err ors in the location of users, th e green cu r ve, can limit the perfo rmance of the propo sal with diminish ing retu rns as the pilot length increa ses, although still significan tly outperfo rming th e b aseline. 12 14 16 18 20 22 24 26 28 30 32 10 -4 10 -3 10 -2 10 -1 10 0 Fig. 6: Performan ce com p arison between the baseline an d the approa c h es with p erfect and imper fect location info rmation for the case with SNR = 0 dB as a fu nction of the number o f users. E. Nu mber of users W e vary the nu mber of users fro m N = 12 to N = 32 , while the r emaining parame te r s stay fixed, and pr esent the results in Fig. 6. The pr oposed m ethod perfo rms exceptionally well in lo wer counts of users, outp erformin g the baseline b y more than fo ur or ders of magnitud e. While ADMM-L I without location errors and strong LoS always outperform s the other approa c h es, it is much more sensitive to the n umber o f users, degrading with a higher rate than the case withou t er rors, and ev en the baseline. M oreover , as th e num ber of users increases, the locatio n erro rs have less impact in the final perf ormance. While it does not prevent perfo rmance from degradin g, using the prop osal even with err ors is pr e ferred over the baseline, as long as they are small. In this ca se, h aving some in f ormation of th e chan nel, is beneficial to the A UD, as it help s to f urther differentiate between users co mpared to o nly when the p ilot sequence is used, e ven if the location is not en tirely correct, as is th e case of th e dar k re d ( µ = 0 . 5 , σ = 0 . 5 ), yellow ( µ = 0 . 5 , σ = 0 ) an d green ( µ → ∞ , σ = 0 . 5 ) curves. F . Numb er of active users W e vary th e number of active users f rom K = 2 to K = 14 and present the results in Fig. 7 . As it is expe c te d in the A UD , as the n umber o f active users increase, the accuracy de creases. In ideal co nditions ( σ = 0 a n d µ → ∞ ), ADM M -LI is the le a st sensiti ve, still fairly o utperfo r ming the b aseline at hig h values of K . Howe ver , even in the presence of weak NL oS ( µ = 5 ), the per formanc e degrades significantly with K , a lthough still outperf orming th e ba selin e. When error s in the lo cation ar e present ( σ 6 = 0 ) , the perf ormance d egrades slo wer with K . V . C O N C L U S I O N S In this p aper we explored one way to use the po sition of u sers to solve the A UD p roblem. Specifically working in the near-field, we prop o sed a metho d that takes into account JOURNAL OF L A T E X CL ASS FILES, VOL. 18, NO. 9, SEPTEMBER 2020 8 2 4 6 8 10 12 14 10 -4 10 -3 10 -2 10 -1 10 0 Fig. 7: Performan ce com p arison between the baseline an d the approa c h es with p erfect and imper fect location info rmation for the case with SNR = 0 dB as a fu nction of the number o f activ e users. prior knowledge of the BS about the users’ localization and the multipath statistics. The p roposed metho d greatly outperf ormed the stand ard approach , though with in creased computatio n complexity of O ( N 6 + N M T ) compar ed to O ( N 3 + N M T ) , and the req uirement o f perf ect knowledge of the users’ position. Ho we ver , we showed th at our method can be used with imper fect loc a lization estimation, still with a significant perfo rmance improvement wh en the po sition error is not to o high and the L oS compon ent is dominant. A P P E N D I X W e minimize (12) by taking the gradient with respec t to each variable at a time and setting it to 0 . For Z , we first define the f unctions g 1 ( Z ) , 1 2 || ΦZ ˆ H − Y || 2 F , g 2 ( Z ) = ρ 2 || X ( s ) − Z + W ( s ) /ρ || 2 F , g 3 ( Z ) , 1 2 tr(( ΦZH )( ΦZH ) H , g 4 ( Z ) , tr(( ΦZ ˆ H ) Y H ) . W e th en take th e gradient of (12) with r espect to Z , and set it to 0 , as ∇ Z L ( X , Z ; W ) = ∇ Z ( g 1 ( Z ) + g 2 ( Z )) = ∇ Z g 1 ( Z ) + ∇ Z g 2 ( Z ) = 0 . (23) Expand ing o n ∇ Z g 1 ( Z ) , we h av e ∇ Z g 1 ( Z ) = ∇ Z  1 2 || ΦZ ˆ H − Y || 2 F  = 1 2 ∇ Z  tr(( ΦZ ˆ H − Y )( Φ Z ˆ H − Y ) H  = 1 2 ∇ Z  tr(( ΦZH )( ΦZH ) H  − ∇ Z (tr(( ΦZH ) Y H )) = 1 2 ( ∇ Z g 3 ( Z ) − ∇ Z g 4 ( Z )) . (24) Then, ∇ Z g 3 ( Z ) is e xpand ed as ∇ Z g 3 ( Z ) = 1 2  tr(( ΦZH )( ΦZH ) H  = 1 2 ∇ Z  tr( Φ H ΦZ ˆ H ˆ H H Z  = 1 2 Φ H ΦZ ˆ H ˆ H H + 1 2 ( Φ H Φ ) H Z ( ˆ H ˆ H H ) H = Φ H ΦZ ˆ H ˆ H H , (25) and for ∇ Z g 4 ( Z ) we h av e ∇ Z g 4 ( Z ) = ∇ Z (tr(( ΦZ ˆ H ) Y H )) = ∇ Z (tr( ˆ HY H ΦZ )) = ( ˆ HY H Φ ) H = ˆ H H YΦ H . (26) Substituting (26) an d (25) in (2 4) we obtain ∇ Z  1 2 || ΦZ ˆ H − Y || 2 F  = Φ H ΦZ ˆ H ˆ H H + ˆ H H YΦ H . (27) Now , ∇ Z g 2 ( Z ) is solved as ∇ Z g 2 ( Z ) = ∇ Z  ρ 2 || X ( s ) − Z + W ( s ) /ρ || 2 F  = − ρ ( X ( s ) − Z + W ( s ) /ρ ) . (28) Finally , with ( 2 7) and (28) we isolate Z as Φ H ΦZ ˆ H ˆ H H + ρ Z = ˆ H H YΦ H + ρ X ( s ) + W ( s ) (29) vec ( Z ) = ( ˆ H ˆ H H ⊗ Φ H Φ + ρ I N 2 ) − 1 vec ( ˆ H H YΦ H + ρ X ( s ) + W ( s ) ) , (30 ) where ⊗ de n otes th e Kro enecker prod uct, and v ec( · ) is the vectorization op eration. Then, Z ( s +1) is obtained by prop erly reshaping vec( Z ) . On the oth er hand, for X , we h ave ∇ X L ( X , Z ; W ) = ∇ X β N X n =1 ν ( r ) n || x n || 2 + ρ 2 || X − Z ( s +1) + W ( s ) /ρ || 2 F ! . T o solve f o r X , we explore the fact that X should only have non-ze r o eleme n ts in its diago nal, simplifying to ∇ X β N X n =1 ν ( r ) n || X n,n || 2 + ρ 2 || X n,n − Z ( s +1) n,n + W ( s ) n,n /ρ || 2 2 ! . (31) By defining d n = Z ( s +1) n,n − W ( s ) n,n /ρ , X can be solved usin g the iterative shrin k age thresholding approa c h [4 0 ] as X n,n = d n | d n | max  0 , | d n | − β ν n ρ  . (3 2) R E F E R E N C E S [1] H . Shariatmada ri et al. , “Machine-t ype Communications: Current Status and Future Perspecti ves T owa rd 5G Syste ms, ” IEEE Commun. Ma g. , vol. 53, no. 9, pp. 10–17, 2015. [2] T . Braud, D. Chatzopoulos, and P . Hui, Machine T ype Communicati ons in 6G . Cha m: Springer Interna tional Publishing, 2021, pp. 207–231. [Online]. 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