Pilot-Based Unsourced Random Access with a Massive MIMO Receiver, MRC and Polar Codes
11 Pilot-Based Unsourced Random Access with aMassive MIMO Receiver, MRC and Polar Codes
Alexander Fengler, Peter Jung, Giuseppe Caire
Abstract
In this work we treat the unsourced random access problem on a Rayleigh block-fading AWGN channel withmultiple receive antennas. Specifically, we consider the slowly fading scenario where the coherence block-lengthis large compared to the number of active users and the message can be transmitted in one coherence block.Unsourced random access refers to a form of grant-free random access where users are considered to be a-prioriindistinguishable and the receiver recovers a list of transmitted messages up to permutation. In this work we showthat, when the coherence block length is large enough, a conventional approach based on the transmission of non-orthogonal pilot sequences with subsequent channel estimation and Maximum-Ratio-Combining (MRC) provides asimple energy-efficient solution whose performance can be well approximated in closed form. For the finite block-length simulations we use a randomly sub-sampled DFT matrix as pilot matrix, a low-complexity approximatemessage passing algorithm for activity detection and a state-of-the-art polar code as single-user error correctioncode with a successive-cancellation-list decoder. These simulations prove the scalability of the presented approachand the quality of the analysis.
Index Terms
Internet of Things (IoT), Unsourced Random Access, Massive Multi-User MIMO, Approximate MessagePassing (AMP).
I. I
NTRODUCTION
Conventional random-access protocols in current mobile communication standards establish an uplinkconnection between user and base station (BS) by first running a multi-stage handshake protocol [1, 2].During this initial access phase active users are identified and subsequently a scheduler assigns orthogonaltransmission resources to the active users. One of the paradigms of modern machine-type communications
The authors are with the Communications and Information Theory Group, Technische Universit¨at Berlin ( { fengler, peter.jung, caire } @tu-berlin.de). a r X i v : . [ c s . I T ] D ec [3] consists of a very large number of devices (here referred to as “users”) with sporadic data. Typicalexamples thereof are Internet-of-Things (IoT) applications, wireless sensors deployed to monitor smartinfrastructure, and wearable biomedical devices [4]. In such scenarios, a BS should be able to collectdata from a large number of devices. However, due to the sporadic nature of the data generation andcommunication, the initial access procedure is overly wasteful.An alternative way of communication is known as grant-free random-access, where users transmit theirdata without awaiting the grant of transmission resources by the BS. A commonly discussed grant-freestrategy in a massive multi-user MIMO setting is to assign fixed orthogonal or non-orthogonal pilotsequences to users [5, 6, 7, 8, 9, 10, 11]. Active users then transmit their pilot sequence directly followedby their data sequence. The BS identifies the active users in the first step and estimates their channelvectors. Subsequently the channel estimates are used to detect the data sequences using either maximum-ratio-combining (MRC) or zero-forcing [6].However, as the number of users in a system grows large and the access frequency becomes small, itgets increasingly inefficient to assign fixed pilot sequences to all the users because even with an efficientsparse recovery algorithm the cost of identifying K a out of K tot users grows with K a log K tot . In contrast, unsourced random access (U-RA) is a novel grant-free paradigm proposed in [12] and motivated by anIoT scenario where millions of cheap devices have their codebook hardwired at the moment of production,and are then disseminated into the environment. In this case, all users make use of the very same codebookand the BS decodes the list of transmitted messages irrespectively of the identity of of the active users.The U-RA approach can simplify the random-access protocol design because it does not required an initialaccess phase and, in contrast to existing grant-free approaches, it allows for a system that is completelyindependent of the inactive users, which makes it well suited to the IoT scenario with a huge number ofdevices with very sporadic activity.Practical coding schemes for U-RA have been mainly studied on the real AWGN channel, e.g. [12, 13,14, 15, 16, 17, 18, 19, 20, 21], and on the Rayleigh quasi-static fading AWGN channel [22, 23]. The U-RAproblem on a Rayleigh block-fading AWGN channel in a massive MU-MIMO setting was formulated in[24] and it was shown that a covariance-based activity detection (AD) algorithm combined with a treecode [17] can achieve sum-spectral-efficiencies that grow proportional to the coherence block-length n ,even if the number of active users is significantly larger then n , specifically up to K a = O ( n ) .In typical wireless systems the coherence block-length n may range from a couple of hundred to acouple of thousand, depending mainly on the speed of the transmitters. At a carrier frequency of GHzthe coherence times, according to the model T c ≈ / (4 D s ) [25], range from ms at km/h to ms at km/h. The coherence bandwidth depends on the maximal delay spread and, in an outdoor environment, typically ranges from to kHz depending on the propagation conditions. Therefore, the numberof complex symbols in an OFDM coherence block may range from n = 100 to n = 20000 , dependingmainly on the assumed speed and the geometry of the environment. Unfortunately, the covariance-basedAD algorithm in [24] has a run-time complexity that scales with n , which makes it unfeasible to useat n (cid:38) . Alternative algorithms based on hybrid-GAMP [26] and tensor-based modulation [27] haveshown an excellent performance at large coherence block-length at which the covariance-based approachwas no longer feasible.In this work we present a conceptually simple algorithm that can be used when n > K a . It is based onpilot transmission, AD, channel estimation, MRC and single-user decoding, very similar to the stat-of-theart approach for massive MIMO grant-free random access [7, 8, 11]. In contrast to the scheme with fixedpilots allocated to all users, we use a pool of non-orthogonal pilots from which active users pick onepseudo-randomly based on the first bits of their message.We show that a collision of users, i.e. two users picking the same pilot sequence, can be resolved byusing a polar single-user code with a successive-cancellation-list (SCL) decoder. Finite-length simulationsshows that the performance of the coding scheme can be well predicted by analytical calculations. Despiteits simplicity the suggested scheme has an energy efficiency that is comparable to existing approaches.Note, that the problem treated here is formally almost equivalent to grant-free random-access with fixedpilots allocated to each user. Differences arise only in the possibility of collisions and the associateduse of an list decodable single-user code. The error probability of AD and MRC in the asymptotic limit K a , K tot , n → ∞ with fixed ratios K a /K tot and K tot /n has been analysed in [8]. In this work we focuson the finite-blocklength regime and the combination of MRC with a single-user polar code.II. C HANNEL MODEL
We consider a quasi-static Rayleigh fading channel with a block of n signal dimensions over which theuser channel vectors are constant. In contrast to the block-fading channel treated in [24], were a messageis encoded over multiple independent fading blocks, here we assume that a message can be transmittedin a single coherence block. Following the problem formulation in [12], each user is given the samecodebook C = { c ( m ) : m ∈ [2 nR ] } , formed by nR codewords c ( m ) ∈ C n . The codewords are normalizedsuch that (cid:107) c (cid:107) ≤ n . An unknown number K a out of K tot total users transmit their message over thecoherence block. Let K a denote the set of active users, i k denote the index of the message chosen byuser k , h k,m ∼ CN (0 , iid be the Rayleigh channel coefficient between user k and receive antenna m and let g k ∈ R + denote the large-scale fading coefficient (LSFC) of user k , which captures the path-loss and shadowing components of the fading. Furthermore, let γ k = g k for k ∈ K a and zero otherwise. Thereceived signal at the m -th receive antenna takes the form y m = K tot (cid:88) k =1 (cid:112) P γ k h k,m c i k + z m = (cid:88) k ∈K a (cid:112) P g k h k,m c i k + z m (1)where z m,i ∼ CN (0 , N ) iid. The BS must then produce a list L of the transmitted messages { m k : k ∈ K a } (i.e., the messages of the active users). The system performance is expressed in terms of the Per-UserProbability of Misdetection , defined as the average fraction of transmitted messages not contained in thelist, i.e., p md = 1 K a E (cid:40) (cid:88) k ∈K a I { m k / ∈L} (cid:41) , (2)In applications a slight overhead in the list size may be tolerable if it reduces the misdetections. Sincethe number of active users is not necessarily known it is practical to let the decoder decide on a list size,which therefore becomes a random variable. Let n fa = |L \ { m k : k ∈ K a }| denote the number of messagesin the output list that were not transmitted by any user, also called False Alarms . n fa is related to the listsize by |L| = n fa + K a (1 − p md ) (3)To get an empirical performance measure we define the Probability of False-Alarm as the average fractionof false alarms, i.e., p fa = E (cid:26) n fa |L| (cid:27) . (4)Operationally, p f a is the probability that a randomly chosen message from the output list is a false alarm.In the special case where |L| = K a is fixed is follows from (3) that p fa = p md .Notice that in this problem formulation the number of total users K tot is completely irrelevant, as longas it is much larger than the range of possible active user set sizes K a (e.g., we may consider K tot = ∞ ).Furthermore, as customary in coded systems, we express energy efficiency in terms of the standard quantity E b /N := PRN .In line with the classical massive MIMO setting [6], we assume an independent Rayleigh fading modelfor the channel coefficients h k,m , such that the channel vectors for different users are independent fromeach other and are spatially white (i.e., uncorrelated along the antennas), that is, h k ∼ CN (0 , I M ) .III. P ILOT - BASED MASSIVE
MIMO U-RALet the coherence block be divided into two periods of lengths n p and n d . In the first period each userschooses one of N = 2 J (non-orthogonal) pilot sequences based on the first J bits of its message. The received signal in the identification phase is given by Y p = AΓ H + Z p ∈ C n p × M , (5)where A ∈ C n p × N is the matrix of pilot sequences with columns normalized as (cid:107) a i (cid:107) = n p , H =( h , ..., h N ) (cid:62) ∈ C N × M is the matrix with the channel vectors as rows and Γ is the matrix with P pilot γ k onthe diagonal. Note, that channel vectors are only defined for those indices which have been chosen by theactive users. Formally, we define the remaining channel vectors as zero. The BS uses an AD algorithm asin [24] to estimate the indices of the used pilots and the corresponding LSFCs. Let ˆ Γ denote the matrixwith the estimated LSFCs on the diagonal and let ˆ I be the estimate of the set of active users. ˆ I can beobtained by either thresholding the estimated received powers, i.e. ˆ I = { k : P ˆ γ k > θ } (6)where θ is some suitable threshold, or, if the number of active users is available, by picking the indicescorresponding to the K a largest estimated received powers. Also for an index set I and for any matrix B let B I denote the matrix that contains only the columns of B with indices in I . Then a linear MMSEestimate of the channel matrix is computed as ˆ H = ˆ Γ / I A H ˆ I (cid:16) A ˆ I ˆ Γ ˆ I A H ˆ I + N I n p (cid:17) − Y p ∈ C ˆ K a × M (7)where A ˆ I denotes a sub-matrix of the pilot matrix A which contains only the columns which have beenestimated as active and ˆ Γ ˆ I contains the LSFCs of the active users on the diagonal. In the second periodeach users encodes its remaining B − J -bit message with a binary ( B − J, n d ) block code and modulatesthe n d coded bits via QPSK on a sequence of n d complex symbols s k . These are transmitted over the n d channel uses in the second phase. The matrix of received signals in the second phase is Y d = (cid:88) k ∈K a (cid:112) P data g k s k h k + Z d ∈ C n d × M . (8)The BS uses the channel estimate ˆ H from the first phase to perform multiuser detection via MRC, i.e. itcomputes ˆ S = ˆ Γ − / I ˆ HY H d ∈ C ˆ K a × n d (9)The rows of ˆ S correspond to estimates of the transmitted sequences s k . Note, that it is also possible touse zero-forcing [6] instead of MRC but this would require that M > K a . The rows of ˆ S are individuallydemodulated, the bit-wise log-likelihood ratios are computed and fed into a soft-input single-user decoder.If the decoder finds a valid codeword, the index of the corresponding pilot is converted back to a J bitsequence and prepended to the codeword. Then the combination of the two is added to the output list. Theuse of a polar code with CRC-bits and a successive-cancellation-list decoder has the additional benefit that we can include all the valid codewords in the output list of the SCL decoder in the U-RA outputlist. This allows to recover the messages of colliding users which have chosen the same pilot in the firstphase. The ability of polar codes to resolve sums of codewords has been observed and used for U-RA onthe AWGN in combination with spreading sequences [20] and a slotted Aloha approach [28, 23]. A. Activity Detection
For the AD in the pilot phase we use the MMV-AMP algorithm, which was introduced in [29], and usedfor AD in a Bayesian setting, where the LSFCs are either known, or its distribution is known in [7, 9]. Thealgorithm aims to recover the unknown matrix X = Γ / H from the linear Gaussian measurements Y p .We restrict the description here to the case where the LSFCs are known at the receiver, the general casecan be found in [7, 9]. Furthermore, we employ some slight modifications here which were introducedin [15]. In the remainder of this section we assume w.l.o.g. that P pilot = 1 . Let X k, : denote the k -th rowof X . Letting λ = K a N be the fraction of active pilots, in the Bayesian setting underlying the MMV-AMPalgorithm it is assumed that the rows of X given g k are mutually statistically independent and identicallydistributed according to p X | g ( x | g k ) = (1 − λ ) δ + λ e − (cid:107) x (cid:107) gk πg k . (10)The MMV-AMP iteration is defined as follows: τ t +1 ,i = (cid:107) Z t : ,i (cid:107) N i = 1 , ..., M (11) X t +1 = η ( A H Z t + X t , τ t +1 ) (12) Z t +1 = Y p − AX t +1 + Nn p Z t (cid:104) η (cid:48) ( A H Z t + X t , τ t +1 ) (cid:105) (13)with X = 0 and Z = Y p . The function η : R M × C N × M → C N × M is defined row-wise as η ( R , τ ) = η ( R , : , τ ) ... η N ( R N, : , τ ) , (14)where each row function η k : R M × C M → C M is chosen as the posterior mean estimate of the randomvector x , with a priori distribution as the rows of X as given above, in the decoupled Gaussian observationmodel r = x + z , (15)where z is an i.i.d. complex Gaussian vector with components ∼ CN (0 , diag ( τ )) . We define η k ( r , τ ) := E [ x | r , g k ] . (16) A simple calculation yields that the function η k ( r , τ ) defined in (16) has the form η k ( r , τ ) = φ t,k ( r , τ ) g k ( g k I M + diag ( τ )) − r , (17)where the coefficient φ k ( r , τ ) ∈ [0 , is the posterior mean estimate of the k -th component b k of theactivity pattern b , when rewriting the decoupled observation model (15) as r = √ g k b k h + z . In particular,we have (details are omitted and can be found in [9]) φ k ( r , τ ) = E [ b k | r , g k ]= p ( b k = 1 | r , g k )= (cid:40) − λλ M (cid:89) i =1 (cid:20) g k + τ i τ i exp (cid:18) − g k | r i | τ i ( g k + τ i ) (cid:19)(cid:21)(cid:41) − (18)The term (cid:104) η (cid:48) ( · , · ) (cid:105) in (13) is defined as (cid:104) η (cid:48) ( R , τ ) (cid:105) = 1 N N (cid:88) k =1 η (cid:48) k ( R k, : , τ ) , (19)where η (cid:48) k ( · , τ ) ∈ C M × M is the Jacobi matrix of the function η k ( · , τ ) evaluated at the k -th row R k, : of thematrix argument R . For the considered model the derivative is explicitly given by η (cid:48) k ( r , τ ) = φ k ( r ) diag ( Ξ k ) + ( Ξ k r )( (cid:101) Ξ k r ) H ( φ k ( r ) − φ k ( r ) ) (20)where we define Ξ k = diag (cid:16) g k g k + τ i : i ∈ [ M ] (cid:17) and (cid:101) Ξ k = diag (cid:16) g k τ i ( g k + τ i ) : i ∈ [ M ] (cid:17) . Calculating the fullmatrix η (cid:48) k ( r , τ ) at each iteration would give a complexity per-row per-iteration of O ( M ) . Note, thatthe diagonal terms are typically much larger then the off-diagonal terms, which is to be expected, sincein expectation the off-diagonal entries of the term (Ξ k r )(˜Ξ k r ) H vanish. Therefore, we approximate thederivative by calculating only the diagonal elements of (Ξ k r )(˜Ξ k r ) H and setting the rest to zero, assuggested in [15].The iterations are repeated for some T max iteration and then an estimate of the LSFCs is obtained as ˆ γ k = φ k ( r kT max , τ T max ) g k , where r kT max is the k -th row of A H Z T max + X T max . B. DFT Pilots and Fast MMV-AMP
To avoid the O ( M n p N ) complexity of the matrix multiplications we create the pilot matrix by choosinga random subset of n p rows of a N × N DFT matrix. This allows to replace the matrix multiplications AX and A H Z in (12) and (13) by FFT operations. Let S = ( s , ..., s n p ) ⊂ [ N ] be a randomly chosensubset. Let W ∈ C N × N be a DFT matrix defined by W i,j = ω jk (21) where ω = e − πi/N . Then we define the pilot matrix A DFT ∈ C n p × N as the submatrix of W with rowsdefined as A DFT i, : = W s i , : . The matrix multiplication A DFT x for some arbitrary vector x ∈ C N is give by ( A DFT x ) i = ( FFT N ( x )) s i (22)for i = 1 , ..., n p , where FFT : C N → C N denotes the FFT operation [30]. For the hermitian transposematrix multiplication A DFT , H z for some vector z ∈ C n p first define ˜ z ∈ C N by ˜ z s i = z i and zero otherwise.Then A DFT , H z = FFT (˜ z ) (23)For the matrix-matrix multiplications A DFT X this process has to be repeated for each column of X leadingto a complexity of O ( M N log N ) . C. Analysis
In this section we calculate an approximate finite-blocklength lower bound on the error probability andon the energy efficiency of the MRC approach. We assume that the AD and LSFC estimation can be donewithout errors, i.e. ˆ Γ = Γ . This gives a lower bound on the error probability and in the regime where n p > K a we expect it to be tight, as in this regime the AD error rates and the error of the LSFC estimationare very low [7, 24]. For simplicity we consider P pilot = P data =: P here. The covariance of the channelestimation error of the LMMSE estimation in (7) is given by C e = I K a − Γ / I A H I (cid:0) A I Γ I A H I + N I n p (cid:1) − A I Γ / I (24)and the MSE of the channel estimate of user k ∈ K a is given by σ k := E {| h k,m − ˆ h k,m | } = ( C e ) k,k . Atypically tight approximation of the effective SINR of each user after MRC is given by [6, 31]
SINR k = (1 − σ k ) g k PN + σ k g k P + (cid:80) K a j =1 ,j (cid:54) = k g j P (25)For orthogonal pilots the channel estimation error reduces to σ k = N N + n p P g k (26)which lower bounds the actual channel estimation error. We investigate this lower bound because theevaluation of (26) is much simpler then (24) since it does not require the inversion of a possibly largematrix. Furthermore, we can evaluate the impact of the non-orthogonality on the channel estimation bycomparing (26) to the true channel estimation errors obtained from (24).An approximation of the achievable rates of a block-code with block-length n d and error probability p e on a real AWGN channel with power SINR is given by the normal approximation [32] R ≈ . SINR ) − (cid:114) V n d Q − ( p e ) (27) where V = SINR SINR + 2(
SINR + 1) log e (28)and Q ( · ) is the Q-function. Using the normal approximation we can find the required SINR to achieve acertain error probability at a given block-length and then we can find the required input power to achievethe target
SINR . D. Simulations
For the simulation in Figure 1 we choose n = 3200 , P e = 0 . , B = 100 , n p = 1152 , n d = 2048 and J = 16 . A randomly sub-sampled DFT matrix is used as pilot matrix and in the AD phase we use theMMV-AMP algorithm with the approximate derivative calculation as described in Sections III-A and III-B.Furthermore, all LSFCs are considered to be constant g k = 1 and known at the receiver. For simplicitywe assume that K a is known at the receiver, and after the MMV-AMP iterations are finished the activecolumns are estimated by picking the K a indices with the largest estimated LSFCs. We use a polar code[33, 34, 35] with a state-of-the-art SCL decoder with CRC bits and a list size of . The simulationsshow a good overlap with the theoretical result. The curves obtained by using formula (26) for the channelestimation error provide a rough but usable approximation which gets worse as K a grows close to n p . Forcomparison we add the reported values of the tensor-based-modulation (TBM) approach [27], with tensorsignature (8,5,5,4,4) and an outer BCH code, although the values have been obtained with the highervalue P e = 0 . . For M = 50 the results pilot based scheme and TBM show a similar shape, althoughthe TBM approach achieves better results for K a ≥ . The results show that with only M = 100 receive antennas over thousand users can be served concurrently, which leads to sum-spectral efficienciesbeyond 30 bits per channel use. This is not surprising, since the scheme essentially resembles pilot basedmassive MU-MIMO with MRC, which is known to achieve very high spectral efficiencies. Nonetheless,the results show that MMV-AMP provides an algorithm that can scale to thousands of concurrent userseven with a large number of non-orthogonal pilots. Furthermore, the combination with a single-user polarcode with list decoding can efficiently reduce the effect of pilot collisions, as we will further investigatein the following. E. Collisions
The average number of collisions of k users on one pilot is given by E { C k } = (cid:0) K a k (cid:1) N k − . (29)We can safely ignore the collisions of more then two users since their number is much smaller than forthe considered parameters. For K a = 1000 and N = 2 (29) gives an average number of − collisions K a -20-15-10-50510 E b / N M= 50M= 100M= 200M= 300Approx.M= 50M= 100M= 200M= 300TBM, M=50, P e = 0.1 Fig. 1: Required energy-per-bit with the MRC approach to achieve P e = p md + p fa < . . n = 3200 and B = 100 are fixed. Solid lines represent the theoretical estimates from Section III-C. Dotted linesrepresent the theoretical results with the assumption of orthogonal pilots. The tensor-based-modulation(TBM) scheme from [27] is included for comparison.of order two. If all of the colliding messages would result in an error, this would lead to a per-user errorprobability of . on average. This can be incorporated into the above analysis by subtracting this valuesfrom the target error probability p e in the normal approximation (27). Nonetheless, if a list decoder isused as a single-user decoder, it is possible to recover both of the colliding messages as demonstrated inFigure 2.In Figure 2 we simulate a collisions between two users on a non-fading AWGN channel and visualizethe probability distribution of the number of correctly decoded messages. We can see that a that at a per-user SNR of about -11 dB the probability that both messages are lost drops below . , where in roughly of the cases both codewords are correctly recovered. The simulation shows that at high SNR valuesthe SCL decoder can reliably recover both messages. When there is no fading, half of the coded bits are − − − − − − − − . . . .
81 SNR [dB] P ( ” N o . o f s u cce ss e s ” = k ) k=0k=1k=2 Fig. 2: Probability distribution of the number of correctly recovered codewords in the case of a two-usercollision in the non-fading case. Polar code with B − J = 84 message bits, n d = 2048 complex QPSKcoded symbols ( real BPSK symbols) and SCL decoding with 16 CRC-bits and list-size 32.erased on average when two codewords are added. Since the rate of the polar code R = ( B − J ) / (2 n d ) ismuch smaller then / , these erasure can be recovered, see also [20]. This situation changes when fadingis involved. In Figure 3 we take the uncertainty of the channel estimation and the MRC into account viathe following simplified two user collision model. Let s and s denote the QPSK modulated sequencesof two users and h , h ∈ C M their iid Rayleigh channel vectors. We model the channel estimates as ˆ h = h + h + e where e ∼ CN (0 , σ est ) is the channel estimation error with variance σ est . The model forthe estimated single-user sequence for both users is then ˆ s (cid:62) = ˆ h H ( √ SNR h s (cid:62) + √ SNR h s (cid:62) + z ) (30)We fix the per-user SNR to -10 dB and vary the variance of the channel estimation error. The simulation inFigure 3 shows that the probability of recovering both codewords saturates to a non-zero value, in contrastto the non-fading case. This effect persist, even when the base per-user
SNR is increased. Nonetheless,the probability that at least one codeword is recovered converges to 1.
F. Complexity
The complexity of the AD with the modified MMV-AMP algorithm is in the order of O ( M N log N ) when the pilots are chosen as the columns of a randomly sub-sampled DFT matrix and the approximatecalculation of the derivatives in the MMV-AMP as described in Sections III-A and III-B. − − − −
10 0 10 20 3000 . . . .
81 1 /σ e [dB] P ( ” N o . o f s u cce ss e s ” = k ) k=0k=1k=2 Fig. 3: Probability distribution of the number of correctly recovered codewords in the case of a two-usercollision with MRC according to the model (30). Polar code with B − J = 84 message bits , n d = 2048 complex QPSK coded symbols and SCL decoding with 16 CRC-bits and list-size 32.R EFERENCES [1] E. Dahlman, S. Parkvall, and J. Skold. . Englisch. 2. Edition. Amsterdam ; New York:Academic Press, Oct. 2013.[2] E. Dahlman, S. Parkvall, and J. Skold.
5G NR: The Next Generation Wireless Access Technology . Englisch. London, United Kingdom; San Diego, CA: Academic Press, Aug. 2018.[3] T. Taleb and A. Kunz. “Machine Type Communications in 3GPP Networks: Potential, Challenges, and Solutions”. en. In:
IEEECommun. Mag.
DOI : 10.1109/MCOM.2012.6163599.[4] M. Hasan, E. Hossain, and D. Niyato. “Random Access for Machine-to-Machine Communication in LTE-Advanced Networks: Issuesand Approaches”. en. In:
IEEE Commun. Mag.
DOI : 10.1109/MCOM.2013.6525600.[5] E. G. Larsson et al. “Massive MIMO for next Generation Wireless Systems”. In:
IEEE Commun. Mag.
DOI : 10.1109/MCOM.2014.6736761.[6] T. L. Marzetta and H. Yang.
Fundamentals of Massive MIMO . en. Cambridge University Press, Nov. 2016.[7] L. Liu and W. Yu. “Massive Connectivity with Massive MIMO-Part I: Device Activity Detection and Channel Estimation”. In:
IEEETrans. Signal Process.
DOI : 10.1109/TSP.2018.2818082. arXiv: 1706.06438.[8] L. Liu and W. Yu. “Massive Connectivity with Massive MIMO-Part II: Achievable Rate Characterization”. In:
IEEE Trans. SignalProcess.
DOI : 10.1109/TSP.2018.2818070. arXiv: 1706.06433.[9] Z. Chen, F. Sohrabi, and W. Yu. “Sparse Activity Detection for Massive Connectivity”. In:
IEEE Trans. Signal Process.
DOI : 10.1109/TSP.2018.2795540. arXiv: 1801.05873.[10] L. Liu et al. “Sparse Signal Processing for Grant-Free Massive Connectivity: A Future Paradigm for Random Access Protocols in theInternet of Things”. In:
IEEE Signal Process. Mag.
DOI : 10.1109/MSP.2018.2844952.[11] K. Senel and E. G. Larsson. “Grant-Free Massive MTC-Enabled Massive MIMO: A Compressive Sensing Approach”. In:
ArXiv180610061Cs Math (June 2018). arXiv: 1806.10061 [cs, math] . [12] Y. Polyanskiy. “A Perspective on Massive Random-Access”. In: .June 2017, pp. 2523–2527. DOI : 10.1109/ISIT.2017.8006984.[13] O. Ordentlich and Y. Polyanskiy. “Low Complexity Schemes for the Random Access Gaussian Channel”. In: (2017), pp. 2533–2537.
DOI : 10.1109/ISIT.2017.8006985.[14] A. Fengler, P. Jung, and G. Caire. “SPARCs and AMP for Unsourced Random Access”. en. In:
IEEE Int. Symp. Inf. Theory Proc. (July 2019), pp. 2843–2847.[15] A. Fengler, P. Jung, and G. Caire. “Unsourced Multiuser Sparse Regression Codes Achieve the Symmetric MAC Capacity”. In: . June 2020, pp. 3001–3006.
DOI : 10.1109/ISIT44484.2020.9174035.[16] R. Calderbank and A. Thompson. “CHIRRUP: A Practical Algorithm for Unsourced Multiple Access”. In:
ArXiv181100879 Eess (Nov. 2018). arXiv: 1811.00879 [eess] .[17] A. Vem et al. “A User-Independent Serial Interference Cancellation Based Coding Scheme for the Unsourced Random AccessGaussian Channel”. en. In: . Kaohsiung, Taiwan: IEEE, Nov. 2017, pp. 121–125.
DOI :10.1109/ITW.2017.8278023.[18] S. S. Kowshik et al. “Short-Packet Low-Power Coded Access for Massive MAC”. en. In: . Pacific Grove, CA, USA: IEEE, Nov. 2019, pp. 827–832.
DOI : 10.1109/IEEECONF44664.2019.9048748.[19] V. K. Amalladinne et al. “On Approximate Message Passing for Unsourced Access with Coded Compressed Sensing”. In:
ArXiv200103705Cs Math (Jan. 2020). arXiv: 2001.03705 [cs, math] .[20] A. K. Pradhan et al. “Polar Coding and Random Spreading for Unsourced Multiple Access”. In: arXiv:1911.01009 (Nov. 2019). arXiv:1911.01009.[21] D. Ustinova et al. “Efficient Concatenated Same Codebook Construction for the Random Access Gaussian MAC”. In: . Sept. 2019, pp. 1–5.
DOI : 10.1109/VTCFall.2019.8891568.[22] S. S. Kowshik et al. “Energy Efficient Coded Random Access for the Wireless Uplink”. In:
IEEE Trans. Commun. (2020), pp. 1–1.
DOI : 10.1109/TCOMM.2020.3000635.[23] K. Andreev, E. Marshakov, and A. Frolov. “A Polar Code Based TIN-SIC Scheme for the Unsourced Random Access in the Quasi-Static Fading MAC”. In:
ArXiv200506899 Cs Math (May 2020). arXiv: 2005.06899 [cs, math] .[24] A. Fengler et al. “Non-Bayesian Activity Detection, Large-Scale Fading Coefficient Estimation, and Unsourced Random Access witha Massive MIMO Receiver”. In:
ArXiv191011266 Cs Math (Aug. 2020). arXiv: 1910.11266 [cs, math] .[25] D. Tse and P. Viswanath.
Fundamentals of Wireless Communication . Englisch. Illustrated Edition. Cambridge, UK ; New York:Cambridge University Press, May 2005.[26] V. Shyianov et al. “Massive Unsourced Random Access Based on Uncoupled Compressive Sensing: Another Blessing of MassiveMIMO”. In:
ArXiv200203044 Cs Eess Math (Feb. 2020). arXiv: 2002.03044 [cs, eess, math] .[27] A. Decurninge, I. Land, and M. Guillaud. “Tensor-Based Modulation for Unsourced Massive Random Access”. In:
ArXiv200606797Cs Math (Aug. 2020). arXiv: 2006.06797 [cs, math] .[28] E. Marshakov et al. “A Polar Code Based Unsourced Random Access for the Gaussian MAC”. In: . Sept. 2019, pp. 1–5.
DOI : 10.1109/VTCFall.2019.8891583.[29] J. Kim et al. “Belief Propagation for Joint Sparse Recovery”. In: arXiv:1102.3289 (Feb. 2011). arXiv: 1102.3289.[30] M. Frigo and S. G. Johnson.
The Fastest Fourier Transform in the West: en. Tech. rep. Fort Belvoir, VA: Defense Technical InformationCenter, Sept. 1997.
DOI : 10.21236/ADA479065.[31] G. Caire. “On the Ergodic Rate Lower Bounds With Applications to Massive MIMO”. In:
IEEE Trans. Wirel. Commun.
DOI : 10.1109/TWC.2018.2808522.[32] Y. Polyanskiy, H. V. Poor, and S. Verd´u. “Channel Coding Rate in the Finite Blocklength Regime”. In:
IEEE Trans. Inf. Theory
DOI : 10.1109/TIT.2010.2043769.[33] E. Arikan. “Channel Polarization: A Method for Constructing Capacity-Achieving Codes for Symmetric Binary-Input MemorylessChannels”. In:
IEEE Trans. Inform. Theory
DOI : 10.1109/TIT.2009.2021379. arXiv: 0807.3917. [34] I. Tal and A. Vardy. “List Decoding of Polar Codes”. In: IEEE Trans. Inf. Theory
DOI : 10.1109/TIT.2015.2410251.[35] A. Balatsoukas-Stimming, M. B. Parizi, and A. Burg. “LLR-Based Successive Cancellation List Decoding of Polar Codes”. In:
IEEETrans. Signal Process.