A Grant-based Random Access Protocol in Extra-Large Massive MIMO System
Otávio Seidi Nishimura, José Carlos Marinello Filho, Taufik Abrão
aa r X i v : . [ ee ss . SP ] S e p A Grant-based Random Access Protocol inExtra-Large Massive MIMO System
Otávio Seidi Nishimura, José Carlos Marinello Filho, Taufik Abrão
Abstract
Extra-large massive multiple-input multiple-output (XL-MIMO) systems is a new concept, wherespatial non-stationarities allow activate a high number of user equipments (UEs). This paper focuseson a grant-based random access (RA) approach in the novel XL-MIMO channel scenarios. Modifi-cations in the classical Strongest User Collision Resolution (SUCRe) protocol have been aggregatedto explore the visibility regions (VRs) overlapping in XL-MIMO. The proposed grant-based RAprotocol takes advantage of this new degree of freedom for improving the number of access attemptsand accepted UEs. As a result, the proposed grant-based protocol for XL-MIMO systems is capableof reducing latency in the pilot allocation step.
Index Terms
Random access protocol, Grant-based, massive MIMO, XL-MIMO, non-stationarity, visibilityregion (VR).
I. I
NTRODUCTION
As stated by the METIS (mobile enablers twenty-twenty society) project [1], there is a predictedrapidly increase in the demand of network access and data traffic for the next few years coming. Toenable such requirement the fifth generation of wireless networks (5G) is expected to provide threemain services: enhanced Mobile Broadband (eMBB), Ultra Reliable Low-Latency Communication(URLLC) and massive Machine Type Communication (mMTC). Another awaited scenario is crowdedMobile Broadband (cMMB), where the number of UEs surpasses those of available pilot sequencesand very high data rate is demanding.
Copyright (c) 2015 IEEE. Personal use of this material is permitted.O. S. Nishimura, T. Abrão are with Electrical Engineering Department, State University of Londrina, PR, Brazil.J. C. Marinello is with Electrical Engineering Department, Federal University of Technology PR, Cornélio Procópio, PR,Brazil. ([email protected], [email protected], taufi[email protected]).This work was supported in part by the Coordination for the Improvement of Higher Education Personnel (CAPES)-Brazil, Finance Code 001, by the Arrangement between the European Commission (ERC) and the Brazilian NationalCouncil of State Funding Agencies (CONFAP), CONFAP-ERC Agreement H2020, by the National Council for Scientificand Technological Development (CNPq) of Brazil under grants 404079/2016-4 and 310681/2019-7.
September 8, 2020 DRAFT
Channel state information (CSI) is necessary to provide coherent communication and this is imple-mented by using orthogonal pilots. However, the number of UEs in crowded scenarios is much greaterthan the available pilot sequences, causing an unfeasible situation to schedule. There are differentmethods of RA, which can be classified in two types: random access to pilots (RAP) and randomaccess to pilots and data transmission (RAPiD) [2]. The second approach is a grant-free RA and usespilot hopping in multiple time slot transmissions, managing pilot collisions and interference withmassive MIMO (mMIMO) properties [3], [4].This paper focuses in RAP, a grant-based RA; herein the transmissions happen in an RA pilotdomain and several UEs are trying to acquire a dedicated pilot for a collision free connection. Apromising protocol to handle many sporadic access attempts is the SUCRe [5]. In general, it resolvesRA pilot collisions, in a totally distributed way, choosing the strongest colliding user and it is wellsettled in a crowded mMIMO system.Since mMIMO is already an essential enabler for 5G networks, in [6] five challenges for thistechnique have been discussed. One of them is to establish how the several conventional mMIMOapproaches will be structured in extra large arrays. These arrays can be implemented under severaltypes of infrastructures, as buildings, stadiums, or shopping malls, where UEs are mainly placed nearthe panels generating non-stationary VRs.The paper contribution consists in proposing a grant-based RA protocol to operate advantageouslyin XL-MIMO systems, in which the large array size and the proximity with the users give rise tospatial non-stationarities across the array. In such configuration, it is possible to take advantage ofUEs distinct VRs as an additional degree of freedom in order to improve the system performancewhile reducing the latency in the pilot allocation step.
Notation : The conjugate, transpose and conjugate-transpose of a matrix A are represented by A ∗ , A T and A H , respectively. I M is the M × M identity matrix, |·| and k·k represent the cardinality of aset and the Euclidean norm of a vector, respectively. Operators E {·} , and V {·} denote the expectationand the variance of a random variable. N ( ., . ) denotes a Gaussian distribution, CN ( ., . ) represents acircularly-symmetric complex Gaussian distribution, and B ( ., . ) represents a binomial distribution. C and R denote spaces of complex and real-valued numbers, while Γ( · ) represents a Gamma function.The operator that gives the real part of its argument is ℜ ( . ) II. S
YSTEM MODEL
For simplicity, our adopted XL-array is a uniform linear array (ULA, Fig. 1), operating in time-division-duplexing (TDD). Since channel modeling is not the focus of this work, it is assumed asimplified bipartite graph model in XL-MIMO, as the one used in [7]. Accordingly, the system isdivided into B subarrays (SAs), each composed by a fixed number of M b = M/B antennas. Let M September 8, 2020 DRAFT be the set composed by , .., B , and V k ⊂ M be the subset of visible SAs associated to user k . Tomodel the VR set V k at random, each SA is independent and identically distributed (i.i.d.) followinga Bernoulli distribution with success probability P b . Then, every UE has a binary vector of size B to indicate if each SA is visible (1) or not (0). For simulation purposes, |V k | > , ∀ k . Fig. 1. Example of a uniform linear extra large array with B = 4 SAs, each with M b = 4 antennas. UEs have differentVRs and consequently distinct associated SAs and channel gains to establish communication. There are K = 12 iUEs, butonly UEs k = 2 , , , want to become active. Let K = U \A be the set of inactive UEs (iUEs), where U is the set of UEs in the entire cell,and A ⊂ U is the subset of active users, each with their dedicated payload pilot. Thus, K = |K| represents the number of iUEs. Let τ p denote the number of mutually orthogonal pilot sequences s , ..., s τ p ∈ C τ p × . In this case, each pilot has length τ p and k s t k = τ p .In this work, it is considered a sliced channel vector h ( b ) k ∈ C M b × between UE k ∈ K and the b -th SA with M b antennas. The vector follows a Rayleigh fading channel model h ( b ) k ∼ CN (0 , β ( b ) k R ( b ) k ) , (1)for all users k = 1 , , ..., K , each with a large scale fading coefficient β ( b ) k . When assuming i.i.d.fading channel, R ( b ) k = I M b , while for correlated fading channels we have [ R ( b ) k ] i,ℓ = r −| ℓ − i | e jθ ( b ) k ( ℓ − i ) , (2)where θ ( b ) k is the angle between k -th UE and the b -th SA, and r ∈ (0; 1) is the correlation index.Actually, a UE has one coefficient per antenna, since the BS is an extra large array. To simplify, β ( b ) k assumes the mean value considering all antennas of SA b , β ( b ) k = M b P M b m =1 β ( b ) k,m , where β ( b ) k,m is thecoefficient between UE k and antenna m ( m = 1 , ..., M b ) at the b -th SA. In addition, invisible SAs September 8, 2020 DRAFT for the k th UE, b / ∈ V k , are assumed to have β ( b ) k = 0 . Moreover, herein, a urban micro scenariomodel [8] is considered: β ( b ) k,m = 10 − κ log( d ( b ) k,m )+ g + ϕ , (3)where d ( b ) k,m represents the distance between UE k and antenna m ( m = 1 , ..., M b ) at the b -th SA, g = − . dB is the pathloss at the reference distance, the pathloss exponent κ = 3 . , and ϕ ∼N (0 , σ ) is the shadow fading, a log-normal random variable with standard deviation σ sf = 10 dB.Each iUE realizes a RA attempt with probability P a ≤ . User k ∈ K uniformly selects an uplinkRA pilot sequence s r ( k ) ∈ C τ p × , where r ( k ) ∈ { , , ..., τ p } . Since transmission is uncoordinated,it is possible and usual that more than one UE choose the same pilot sequence s t . Therefore, let S t = { k : r ( k ) = t, ρ k > } represent the set of iUEs indices transmitting pilot t , with power ρ k .The cardinality of this set follows a binomial distribution [5]: |S t | ∼ B (cid:18) K, P a τ p (cid:19) . (4)Fig. 2 depicts an arbitrary uplink RA arrangement with K = 3 , B = 4 , τ p = 1 and P a = 1 . In thiscase, there are collisions in SAs 1, 3 and 4 between users 1 and 2, and 2 and 3, but no collisionsbetween users 1 and 3. Fig. 2. An example of the proposed UL arrangement with a probability P a = 1 , K = 3 users, B = 4 subarrays and τ p = 1 available pilot sequence. SUCRe protocol relies on mMIMO properties, as channel hardening and asymptotic favorablepropagation: k h ( b ) k k M b M b →∞ −−−−−→ β ( b ) k , ∀ k, b (5) h ( b ) Hk h ( b ′ ) k ′ M b M b →∞ −−−−−→ , ∀ ( k, b ) = ( k ′ , b ′ ) , (6)respectively. From eq. (5), it follows that X j ∈V k k h ( j ) k k M b M b →∞ −−−−−→ X j ∈V k β ( j ) k , ∀ k (7) September 8, 2020 DRAFT which represents the overall channel gain over the visible SAs for k -th UE. Notice that the numberof antennas per SA, M b , does not always remain large, since VRs represent just a portion of antennasavailable for each user in a specific time. Nevertheless, the proposed protocol, named SUCRe-XL,still presents a satisfying performance even under certain reduced number of antennas per SA.III. P ROPOSED
SUCR E -XL PROTOCOL
We first describe how a straightforward adaptation of the conventional SUCRe protocol to the XL-MIMO scenario would be, demonstrating why it does not work. Then, we propose the SUCRe-XLprotocol deploying modifications to operate in the XL-MIMO regime in step 2. The section concludeswith the definition of the contention resolution rules and allocation strategy for the dedicated payloadpilots.
Step 1 : Random UL Pilot Sequence . All UEs that want to be active send RA pilot sequences. In theBS, the b th SA receives signal Y ( b ) ∈ C M b × τ p : Y ( b ) = X k ∈K √ ρ k h ( b ) k s Tr ( k ) + N ( b ) , (8)where N ( b ) ∈ C M b × τ p is the receiver noise, with i.i.d. elements distributed as CN (0 , σ ) . To estimatethe channel of UEs k ∈ S t ( t = 1 , ..., τ p ), the BS correlates Y ( b ) for each sub-array b with eachnormalized pilot sequence s t , y ( b ) t = Y ( b ) s ∗ t k s t k = X i ∈S t √ ρ i τ p h ( b ) i + n t , b = 1 , . . . , B. (9)where n t = N s ∗ t k s t k ∼ CN (0 , σ I M b ) is the effective receiver noise. With eq. (5) and (6), the followingapproximation holds: k P b ∈M y ( b ) t k M b M b →∞ −−−−−→ X b ∈M X i ∈S t ρ i β ( b ) i τ p | {z } α t + Bσ . (10)The proof of property (10) is found in the appendix. Hence, the sum of the signal gains , α t , receivedat the BS for each RA pilot in step 1 is readily identified as the first term in (10). Step 2 : Precoded Random Access DL Response . In the second step of the SUCRe procedure, eachSA responds with an orthogonal precoded DL pilot V ( b ) ∈ C M b × τ p . Using a normalized conjugateof y ( b ) t , results: V ( b ) = r qB τ p X t =1 y ( b ) ∗ t k y ( b ) t k s Ht , b = 1 , . . . , B, (11)where q is the predefined DL transmit power. Then, UE k ∈ S t receives signal v Tk ∈ C × τ p given by v Tk = X m ∈V k h ( m ) Tk V ( m ) + η Tk , (12) September 8, 2020 DRAFT where η k ∼ CN (0 , σ I τ p ) is the receiver noise. Next, each UE correlates the received signal in eq.(12) with RA pilot s t : v k = v Tk s t k s t k = r qτ p B X m ∈V k h ( m ) Tk y ( m ) ∗ t k y ( m ) t k + η k , (13)where η k = η Tk s t k s t k ∼ CN (0 , σ ) is the effective receiver noise. Dividing the equation by √ M b , andconsidering that asymptotic conditions of eq. (5) and (6) hold, it follows that: v k √ M b = r qτ p B X m ∈V k (cid:16) h ( m ) Hk y ( m ) t (cid:17) ∗ M b r M b (cid:13)(cid:13)(cid:13) y ( m ) t (cid:13)(cid:13)(cid:13) + η k √ M b (14) M b →∞ −−−−−→ X m ∈V k p ρ k q/Bτ p β ( m ) k r P i ∈S t ρ i β ( m ) i τ p + σ . Notice that the magnitude α t received at the BS, as in eq. (10), cannot be mathematically separated,due to the sum of different denominators. Since the users cannot obtain this information, the appli-cation of the strongest user criterion becomes difficult. For this reason, the following SUCRe forXL-MIMO protocol is proposed. SUCRe-XL Precoded DL Response . In the second step of the SUCRe-XL protocol, instead of em-ploying conjugate- y ( b ) t precoding as in eq. (11), all SAs use the same precoding vector P b ∈M y ( b ) t .Thus, each SA responds with the same signal V XL ∈ C M b × τ p : V XL = r qB τ p X t =1 P b ∈M y ( b ) ∗ t k P b ∈M y ( b ) t k s Ht . (15)Then, the UE k ∈ S t receives signal z Tk ∈ C × τ p , z Tk = X m ∈V k h ( m ) Tk V XL + η Tk , (16)and correlates it with RA pilot s t : z k = z Tk s t k s t k = r qτ p B X m ∈V k h ( m ) Tk P b ∈M y ( b ) ∗ t k P b ∈M y ( b ) t k + η k . (17)In the same way of eq. (14), it follows that: September 8, 2020 DRAFT z k √ M b = r qτ p B P m ∈V k h ( m ) Hk P b ∈M y ( b ) t ! ∗ M b s M b (cid:13)(cid:13)(cid:13)(cid:13) P b ∈M y ( b ) t (cid:13)(cid:13)(cid:13)(cid:13) + η k √ M bM b →∞ −−−−−→ p ρ k q/Bτ p P m ∈V k β ( m ) k r P b ∈M P i ∈S t ρ i β ( b ) i τ p + Bσ . Thus, noise and estimation errors in the imaginary part are removed from eq. (17), resulting ℜ ( z k ) √ M b ≈ p ρ k q/B P m ∈V k β ( m ) k τ p p α t + Bσ . (18)Hence, the k th UE can now have an estimate by isolating α t . The estimator of [5] can be readilyadapted to our RA XL-MIMO scenario as b α t,k = max " ρ k X m ∈V k β ( m ) k τ p , (19) (cid:18) Γ( M b + 1 / M b ) (cid:19) ρ k qτ p (cid:16)P m ∈V k β ( m ) k (cid:17) B [ ℜ ( z k )] − Bσ . It is proved that changing the precoding as in eq. (15) and adapting the b α t,k estimator as in eq. (19)are sufficient to implement the proposed RA protocol in XL-MIMO scenarios. Such procedure doesnot cause any additional overhead or sum rate loss in comparison with the original SUCRe protocol[5]. Step 3 : Contention Resolution and Pilot Repetition . To resolve contentions distributively and unco-ordinately, the k -th UE now has b α t,k , which is the summation of the contending UEs signal gainswith its own ρ k P m ∈V k β ( m ) k τ p . However, the number of contenders |S t | as well as the VRs of eachUE are unknown by the users, leading to the only possibility of comparing its own overall gain with b α t,k , by computing ρ k b α t,k P m ∈V k β ( m ) k τ p . Hence, UEs using the SUCRe-XL protocol apply the following decision rule : R k : X m ∈V k ρ k β ( m ) k τ p > b α t,k / ǫ k (repeat) , (20) I k : X m ∈V k ρ k β ( m ) k τ p ≤ b α t,k / ǫ k (inactive) . (21)In this decision rule, the bias term ǫ k is given by ǫ k = δ √ M b X b ∈V k β ( b ) k (22) September 8, 2020 DRAFT where δ is an adjustable scale factor for finding a suitable operation point. As in [5], we adopt a δ = − .There are four possible cases in a contention process: i . Non-overlapping UEs win (false positive).Ex.: from Fig. 2 users 1 and 3 win. ii . Only one UE wins. iii . None of the UEs win (false negative). iv . Overlapping UEs win (false positive). Ex.: from Fig. 2, users 1 and 2 or 2 and 3 win. Althoughcase 1 is a false positive, there is no pilot collision. Therefore, cases 1 and 2 are successful attemptsand there is the allocation of the RA pilot. Case 4 is considered a pilot collision; i.e. , a pilot collisionoccurs if more than one UE in S t retransmit in step 3 and have overlapping VRs. Step 4 : Allocation of Dedicated Payload Pilots
After the BS receives the repeated UL pilot trans-missions from step 3, it tries to decode the message with new channel estimates from the repeatedpilots. If the decoding goes well, the BS can allocate pilot sequences in the payload data blocks tothe non-overlapping contention winners, followed by a replying DL message informing the successfulconnection and, if necessary, more information. If the decoding fails, the protocol failed to resolvethat collision and the unsuccessful UE is instructed to try again after a random interval.
SUCRe-XL Complexity is equivalent to that of conventional SUCRe protocol. Although the computa-tion of the precoding vector increases marginally at the BS with the number of SAs B , due to the sumof all different estimated channels in (15), the same precoding vector is used for all SAs, differentthan the precoding in (11) for the original SUCRe. While the original SUCRe has to compute B different vector inner products in (11), the proposed SUCRe-XL protocol has to compute a sum of B vectors followed by a single vector inner product in (15). Also, each UE k ∈ S t , ∀ t has to estimatethe sum of its large scale fading coefficients in SUCRe-XL protocol, which can be evaluated as theaverage received power of a beacon signal in a step 0, similarly as assumed in [5].IV. N UMERICAL R ESULTS
It is assumed a 100 meter ULA with M = 500 antennas in a × m square cell with K = 1000 uniformly distributed iUEs (crowded scenario) as illustrated in Fig. 1, each user wantsto become active with probability P a = 0 . . It is considered τ p = 10 pilots, and transmit powers ρ k = q = 1 W, ∀ k . Two channel models were deployed: i ) uncorrelated Rayleigh fading, as in eq.(1), with R ( b ) k = I M b ; ii ) correlated Rayleigh fading model, following eq. (2), with r = 0 . .A baseline ALOHA-like performance has been included for comparison purpose, which treats pilotcollision by retransmission after a random waiting time period, hence, contending users retransmittheir pilots at random if collision occurs.The Probability to Resolve Collision (PRC) is calculated numerically taking all resolved collisionsper total number of collisions occurred. Simulations were carried out in sequential RA blocks fashion,where iUEs try to access the channel in each iteration. For each parameter value of the x -axis ( B or September 8, 2020 DRAFT P b in Fig. 3), it is simulated sequential RA blocks. If an attempt fails, UE makes another attemptwith probability 0.5 in the subsequent blocks. It is given a limit of 10 RA attempts per UE, afterwhich a failed access attempt is declared.Fig. 3(a) depicts the PRC and the normalized mean square error (NMSE), given by E {| b α t,k − α t | } /α t . It shows that increasing the number of SAs B , which means reducing the number ofantennas per SA M b , since M/B = M b , causes a progressive discrepancy on α t estimation dueto (5) and (6) do not hold when M b decreases. Indeed, NMSE levels for the SUCRe-XL protocoldeteriorate steadily when B > for both channel models. To simplify this simulation, |V k | = B, ∀ k .The PRC starts increasing until B = 25 for the uncorrelated Rayleigh fading, and presents an optimalPRC value when B = 10 for the correlated Rayleigh fading model . B = 1 corresponds to a spatialstationary regime.
10 20 30 40 50
Number of Subarrays (B) -3 -2 -1 N o r m a li z ed M ean S qua r ed E rr o r NMSE: CorrelatedNMSE: UncorrelatedPRC: BaselinePRC: SUCRe-XL CorrPRC: SUCRe-XL Uncorr P b P r obab ili t y t o R e s o l v e C o lli s i on SUCRe-XL (B = 1)SUCRe-XL (B = 5)SUCRe-XL (B = 20)Baseline (B = 1)Baseline (B = 5)Baseline (B = 20) (a) (b)
Fig. 3. (a)
Left y-axis : NMSE to verify the α t estimation. Right y-axis : Probability to resolve collision (PRC) againstthe number of subarrays B . Baseline performance remains the same for both channel models. (b) PRC vs. P b for theuncorrelated Rayleigh model. Fig. 3(b) depicts the PRC for different probabilities of each SA being visible for a given UE, P b . Notice that P b is inversely proportional to the density of obstacles affecting transmitted signals.The probability of the VRs of |S t | UEs in (4) not overlapping, given by P no = ((1 − P b ) |S t | + The initial PRC increase is due to the SUCRe-XL decision rule associated with the possibility of users retransmittingthe same RA pilot having non overlapping VRs, but then the reduced number of antennas per SA diminishes the channelhardening and favourable propagation effects, as well as the quality of the b α t,k estimates and, consequently, the PRC.Channel correlation highlights this effect, making the PRC starts to decrease with a lower B value. September 8, 2020 DRAFT0 |S t | P b (1 − P b ) |S t |− ) B , decreases with increasing B and/or P b . Thus, decreasing P b causes |V k | ∀ k to diminish at the BS side. Hence, the probability of pilot collisions in overlapping areas reduceswhen probability P b decreases, improving the SUCRe-XL PRC for B = 20 , as well for the Baselineperformance. However, this could not be seen for the SUCRe-XL for B = 5 , since the effect of theimposed constraint |V k | ≥ is more noticeable when P b ≤ /B . Thus, when decreasing P b belowthe threshold /B , the expected value of visible subarrays would decrease below 1, in such a waythat the additional constraint |V k | ≥ turns to intervene more frequently, breaking the trend of thepresented result in increasing the PRC with the decrease of P b , as expected according to the P no expression. Besides, this does not occur for B = 1 (stationary case), since the only SA existent is theentire linear array. Furthermore, the Baseline success probability grows abruptly comparing with theproposed protocol with P b reduction. This behavior might come from non-overlapping cases, whenthe decision rule would be unnecessary: the Baseline recognizes non-overlapping pilot collisions assuccessful attempts, while UEs in the SUCRe-XL protocol still have to decide to repeat the RA pilot,even when they are not overlapping. Average Number of Access Attempts . Numerical results in Fig. 4(a) shows the average number of RAattempts as a function of the number of iUEs. The fraction of UEs that could not access the network, i.e. , the portion that is unsuccessful in the maximum number of 10 RA attempts, is illustrated in Fig.4(b). There is a clear advantage of SUCRe-XL in reducing the failed access attempts when exploitingthe channel non-stationarities, supporting a higher number of UEs.Fig. 5(a) depicts the average number of accepted UEs per resolved collision ( ξ ), showing that ξ remains around one with increasing number of subarrays. Although ξ is slightly higher for theBaseline scheme, the resolved collisions are much rarer in this simple scheme, as in Fig. 3(a). Inthe same scenario, Fig. 5(b) indicates the normalized number of accepted UEs ( λ ) that realizedsuccessful attempts. Hence, in average, the total number of admitted UEs along the RA blocksis given by
Λ = λ · K · P a · . Indeed, non-stationary cases surpasses the stationary one, speciallyin (over)crowded mMTC scenarios, being able to manage a greater number of UEs.V. C ONCLUSION
Grant-based RA operating under massive antennas has demonstrated satisfactory performance tohandle multiple access attempts under (over)crowded scenarios, typically present in cMMB. Further-more, XL-MIMO is a promising concept to surpass the performance of classical antenna structures.Hence, to take advantage of channel non-stationarities, an adapted SUCRe protocol for XL-MIMO To avoid the possibility of a given user do not see any subarray, while the average number of visible subarrays per userfollows E [ |V k | ] = B · P b . September 8, 2020 DRAFT1
Number of Inactive UEs (K) A v e r age N u m be r o f A cc e ss A tt e m p t s Number of Inactive UEs (K) F r a c t i on o f F a il ed A cc e ss A tt e m p t s SUCRe-XL (B = 1)SUCRe-XL (B = 5)SUCRe-XL (B = 20)Baseline (B = 1)Baseline (B = 5)Baseline (B = 20) (b)(a)
Fig. 4. SUCRe-XL and Baseline performance in the uncorrelated Rayleigh fading model ( P b = 0 . ). (a) Average numberof RA attempts. (b) Probability of failed access attempts. Number of Subarrays (B)
SUCRe-XLBaseline
Number of Inactive UEs (K)
SUCRe-XL (B = 1)SUCRe-XL (B = 5)SUCRe-XL (B = 20)Baseline (B = 1)Baseline (B = 5)Baseline (B = 20) (b)(a)
Fig. 5. SUCRe-XL and Baseline performance in the uncorrelated Rayleigh fading model. (a) Average number of UEsper resolved collision vs.
B. (b) Normalized number of accepted UEs for different numbers of iUEs, considering RAblocks.
September 8, 2020 DRAFT2 has been proposed and compared. Besides, the proposed protocol can support a higher number ofactive UEs, since it attains a reduced fraction of failed access attempts and reduces access latency.A
PPENDIX
For simplicity, let ρ i be the same for all i ; then we have: X b ∈M y ( b ) t ! H = X b ∈M X i ∈ S t √ ρ i τ p h ( b ) Hi + n ( b ) Ht . (23)Then, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X b ∈M y ( b ) t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = X b ∈M y ( b ) t ! H · X b ∈M y ( b ) t ! = ρ i τ p X b ∈M X i ∈ S t k h ( b ) i k + 2 ρ i τ p X b ∈M X i,j ∈ S t i = j h ( b ) Hi h ( b ) j ++ 2 ρ i τ p X m,b ∈M m = b X i,j ∈ S t h ( m ) Hi h ( b ) j + 2 √ ρ i τ p X m,b ∈M X i ∈ S t h ( m ) Hi n ( b ) t + 2 X m,b ∈M m = b n ( m ) Ht n ( b ) t + X b ∈M k n ( b ) t k Dividing (cid:13)(cid:13)(cid:13)P b ∈M y ( b ) t (cid:13)(cid:13)(cid:13) by M b → ∞ , components with different indices, as the second to the fifth,become zero, following property in eq. (6). Furthermore, the first component obeys approximation(5), resulting in β i . The last term becomes noise variance, validating approximation (10).R EFERENCES [1] M. Fallgren and B. T. et al, “Deliverable d1.1: Scenarios, requirements and kpis for 5G mobile and wireless system,”no. ICT-317669-METIS, 2013.[2] E. d. Carvalho, E. Bjornson, J. H. Sorensen, P. Popovski, E. G. Larsson, “Random access protocols for massive mimo,”
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