I Introduction
Ia Motivation
One imminent demand for the next generation wireless mobile communication systems is to provide instant and reliable access for an increasingly large number of machinetype devices (MTDs) [bockelmann2016massive, shariatmadari2015machine]. Different from humancentric communication, the resultant Massive MachineType Communication (mMTC) has two distinct features. In particular, only a small number of devices are active in each communication round due to the sporadic activity in mMTC[chen2018sparse]. Besides, MTDs usually transmit small data payloads adopting shortpacket signaling [durisi2016toward]. These make traditional grantbased random access schemes generally not very suitable for the mMTC scenario because of their low spectral efficiency and exceedingly long latency [chen2020massive]. Therefore, the design of reliable and efficient grantfree random access schemes has attracted significant attention recently, where active users transmit pilots and data to the base station (BS) directly without permission granted [liang2017non, zhang2016grant]
. In most grantfree random access schemes, a set of pilot sequences that are designated to the users are used for the BS to ensure its accurate user activity detection and channel estimation
[senel2018grant, kim2019novel, wang2019joint]. However, this is neither affordable nor feasible in the next generation multiple access (NGMA) scenarios due to the high density, the large number of connections therein, and the frequent collisions that may occur. To tackle the issues, a special type of grantfree random access, the socalled unsourced random access (URA), is introduced in [polyanskiy2017perspective], in which users do not transmit preambles, all the potential users share a common codebook, and the BS only needs to decode a list of messages instead of the identities of active users. This scheme can avoid the huge cost of preambles and the extra protocol of collision resolution, thus well meeting the requirements of next generation massive access.On the other hand, massive or super MIMO technology, in combination with the millimeterwave (mmWave) technology, have been promoted as two core technological features for the next generation wireless communication system with a witnessed potential to boost the capacity and efficiency. These two underlying key technologies jointly bring additional spatialdomain signal dimension with their excellent intrinsic directivity and proper beamforming, and also result in salient beamspace sparsity due to the lack of scattering in a mmWave MIMO channel [akdeniz2014millimeter, wen2014channel, bellili2019generalized]. To further increase the efficiency of the future massive access systems, such spatialdomain resources and properties should be fully explored and exploited. Various multiuser transmission schemes have been proposed to unleash the potential and properties of the beamspace resources, such as the typical works on beam division multiple access (BDMA), which simultaneously serves multiple users via different beams [sun2015beam, you2017bdma, jia2019massive].
However, for grantfree random access, even if the information of the location of all potential users is stored at the BS, the identities of active users are unknown to the BS. Therefore, the beam dimension cannot be directly exploited in the design of the encoding process of an unsourced grantfree random access, which is different from previous works [sun2015beam, you2017bdma, jia2019massive]. Moreover, in a general URA system, the messages of active users are divided into several subblocks and transmitted in consecutive subslots. As the signals transmitted from active users often experience deep fading, some subblocks may be missed by the decoder at the receiver. The loss of any subblock in any subslot finally leads to the failure of recovering the corresponding original message. The problem of packet loss is also needed to be solved as it can cause severe decoding performance degradation. Note that the user location information can generally serve as a hint of the indices of the messages originated from it. Therefore, the intrinsic beam division property and salient spatial sparsity in a mmWave MIMO system can provide extra extrinsic information for both multiuser signal separation and multisubblock message stitching. This motivates us to exploit the beam space properties to design new URA schemes for next generation multiple access to help the entire system accommodate more active users and improve the decoding performance.
IB Related Works
Y. Polyanskiy first introduced a framework named URA [polyanskiy2017perspective]. Specifically, in URA, all the users share a common codebook, and the decoder only needs to decode a list of messages transmitted from the active users. The error probability is defined as the average fraction of misdecoded messages over the number of active users, including both missed detection and false alarm. It is obvious that the message recovery at the BS can be formulated as a compressed sensing (CS) problem due to the sporadic activity in mMTC, which is similar to the conventional grantfree random access schemes [liu2018massive, liu2018massive2]. However, the size of the common codebook grows exponentially with the number of information bits. In practice, even if a short packet is transmitted, the size of information messages is typically at the order of 100 bits, which makes the CS algorithms computationally intractable. In this context, V. K. Amalladinne proposed a coded compressed sensing (CCS) scheme for URA communication [amalladinne2020coded]. In particular, the messages from active users are first divided into several subblocks. Then, a systematic linear code adds redundancy to those subblocks. Once this is achieved, each subblock is mapped into a codeword in a common codebook and transmitted in a certain subslot. Then a standard CS algorithm implements the detection of the subblocks. Finally, the subblocks transmitted in different subslots are stitched together to obtain the original messages. Build upon the findings in [amalladinne2020coded] and the structure of sparse regression codes (SPARCs), A. Fengler provided an improved inner decoder, and a complete asymptotic error analysis [fengler2021sparcs].
Apart from the above works, the study of massive multipleinput multipleoutput (MIMO) URA has also attracted much attention. A. Fengler extended the URA model of [amalladinne2020coded] to a blockfading MIMO channel by using a lowcomplexity covariancebased CS (CBCS) recovery algorithm [fengler2021non]. Considering the low code rate and spectral efficiency of the CCS scheme, V. Shyianov proposed a new algorithmic solution to solve the massive URA problem by leveraging the rich spatial dimensionality offered by largescale antenna arrays [shyianov2020massive]. Besides, without requiring a separate activity detection or channel estimation step, A. Decurninge
introduced a structure that allows the receiver to separate the users using a classical tensor decomposition
[decurninge2020tensor]. As URA is a special scheme of grantfree random access, A. Fengler presented a conceptually simple algorithm based on pilot transmission, activity detection, channel estimation, Maximum Ratio Combining (MRC), and singleuser decoding [fengler2020pilot], which is similar to the existing grantfree random access schemes [chen2018sparse, liu2018massive]. The difference is that they use a pool of nonorthogonal pilots where every active user picks one of them pseudorandomly. Furthermore, X. Shao proposed a unified cooperative activity detection framework for sourced and unsourced random access based on the covariance of the received signals for the sixth generation (6G) cellfree wireless networks [CAD].IC Main Contributions
In this paper, we propose a URA scheme with beamspace tree decoding. Specifically, we adopt the CCS scheme [amalladinne2020coded] suitably to our case and design two beamspace tree decoders, which are based on hard decision and soft decision, respectively. By leveraging the beam division property to assist in distinguishing the subblocks transmitted from different users, both decoders can help the system serve more active users. As the discriminating power is improved, the searching space of the solution in the decoding process is reduced, such that the first decoder has low complexity. In addition, notice that any subblock missed by the CS decoder would finally lead to missed detection, which degrades the decoding performance. To tackle this issue, the second decoder establishes factor graphs at each stage during the decoding process and implements message passing algorithm (MPA) to give each candidate subblock drawn from the checking relationship a loglikelihood ratio (LLR) value. Then the reliability of every candidate path is calculated by a path metric (PM). At every stage, some reliable paths are kept, and finally, the surviving path is output as the valid message. Even if a subblock is missed by the CS decoder, it is possible that the path of the original message is reliable and kept. The main contributions of this paper are summarized as follows:

A URA scheme with beamspace tree decoding is proposed for mmWave communication systems in mMTC to accommodate more active users and to improve the system performance.

Two beamspace tree decoders are designed. Both of them can exploit the intrinsic beam division property to improve the decoding performance of the tree decoder by enhancing the discriminating power and helping the system serve more active users. Besides, the first decoder is based on hard decision with low complexity. The second one is based on soft decision and exploits the advantage of list decoding to recover the packet loss, which is the key of the proposed URA scheme.

Simulation results verify that our URA schemes have significantly better performances than existing works.
ID Paper Organization and Notations
The rest of this paper is organized as follows: Section II provides a brief introduction of the considered massive URA system. Section III provides the encoding and decoding process of the considered system. Section IV proposes a beamspace tree decoder with hard decision. Then, Section V designs a beamspace tree decoder with soft decision. Next, Section VI analyzes the performance of the proposed URA scheme. Afterward, Section VII provides extensive simulation results to validate the effectiveness of the proposed algorithm. Finally, Section VIII concludes the paper.
Throughout this paper, we use bold letters to denote matrices or vectors and nonbold letters to denote scalars. We denote the
th row and the th column of a matrix with the rowvector and the columnvector respectively. We denote by the space of complex matrices of size . We use to denote the absolute value of a complex number, and to denote conjugate transpose and transpose, respectively. The norm of an input vector is denoted by . denotes the number of elements of set . The notationdenotes that the random variable (r.v.)
follows the circular symmetric complex Gaussian distribution.
stands for the bigO notation.Ii System Model
Consider an uplink singlecell cellular network consisting of singleantenna users. The BS is equipped with antennas and radio frequency (RF) chains such that , as shown in Fig. 1. Due to the sporadic user activity of mMTC, only a small number of users are active in a transmisson process, i.e., . Each active user has bits of information to be transmitted in a blockfading channel. According to [akdeniz2014millimeter, bellili2019generalized], the channel vector from user to the BS can be written as
(1) 
where denotes the total number of clusters and within the th cluster there are subpaths. and denote the gain and the the angle of arrival (AOA) of the th subpath within the th cluster. For the uniform linear array (ULA), the array steering vector can be expressed as
(2) 
where , , is the signal wavelength, and is the antenna spacing which is usually half of the signal wavelength.
To overcome the strong path loss in mmWave channels, a beamforming technique should be adopted. However, the BS cannot focus in any specific direction in grantfree random access. The reason is that even if the location of all potential users is stored at the BS, which users are active is not prior information known to the BS. Besides, due to the constraint of hardware implementations and large energy consumption of RF chains, we have . Therefore, many beamforming methods in existing works [gao2016near, wang2017spectrum] cannot be applied in our system directly as
narrow beams cannot cover the whole beam space. In this paper, we give a beamforming method based on the widely used Discrete Fourier Transform (DFT) based beamforming codebook
[gao2016near, wang2017spectrum], to overcome the strong path loss of mmWave channels in grantfree random access. Specifically, the DFT based beamforming codework, which is denoted by , can be writtern as(3) 
where
(4)  
Consider the process of hardware implementations, the number of antennas is usually a multiple of the number of RF chains. Therefore, the consecutive beamforming vectors can be grouped and summed together to form a new beamforming vector , . is expressed as
(5) 
where the parameter is set to constrain the power of receive beamforming, i.e., . Then the beamforming matrix is obtained, where is written as . By applying this beamforming method, the width of every beam is , thus the beams can cover the whole beam space, which means that the signals coming from all directions can be received by the BS.
In a typical URA scenario, all the users share a common codebook , which is denoted by . The power of each codeword is constrained to 1, i.e., . Let denote whether user transmits the codeword . can be written as
(6) 
After receive beamforming at the receiver, the beam domain channel vector of the active user is denoted by . Also, the random noise vector is denoted by , where is modeled by a complex circular Gaussian random vector with i.i.d. components, i.e., . Then the received signal on the th beam can be written as
(7) 
By summarizing all the samples in a transmission block, the received signal can be recast as
(8) 
where , , and . The matrix contains only nonzero columns each of which having a nonzero entry.
For the matrix , where , the th row of such matrix is given as
(9) 
The probability that is identically zero is given by . Since is significantly larger than , the matrix is rowsparse, which is shown in Fig. 2. The reason is that only a small number of users are active due to the sporadic traffic of users, i.e., . For the same reason, the matrix is also rowsparse. Moreover, due to the lack of scattering in mmWave bands, the signal propagates from the transmitter to the receiver through a small number of path clusters. This leads to the sparsity of mmWave massive MIMO channels in the beam domain as well, i.e., the channel vector is sparse. Therefore, for the matrix , the number of nonzero entries of its columns is less than that of , which is shown in Fig. 2. Note that the total number of users plays no role in the matrix . This means that if the matrix is recovered, only the codeword index is known to the BS, instead of the user’s ID, which leads to the socalled unsourced property.
Let and denote the set of the recovered messages at the BS and the set of the active users, respectively. Each active user expects to transmit bits of information, i.e., . The performance in URA is evaluated by the probability of missed detection and false alarm, denoted by and respectively, which can be given by:
(10) 
(11) 
and the error probability of the system is defined as
(12) 
Iii Proposed URA Scheme
In this section, we first review the studies of the CCS scheme in [amalladinne2020coded] and then propose a URA scheme. In the CCS scheme, each active user partitions the message into several subblocks and adds parity bits. The CS techniques detect the subblocks transmitted by active users in all subslots. A treebased algorithm then stitched these subblocks to recover the original messages.
Iiia Encoding Process
The transmission strategy includes two encoders: tree encoder and CS encoder. The tree encoder uses a systematic linear block code based on random parity checks to add parity bits to every subblock. The CS encoder maps each subblock into a codeword in the common codebook.
IiiA1 Tree Encoder
Divide bits message into subblocks of size , where . Let and , . Each subblock is resized to length by appending parity bits, which is obtained by linear combinations of the information bits of the previous subblocks. Mathematically, define as a coded message, then we have . Herein, is obtained by
(13) 
where is a binary matrix. Parity bits are computed using modulo2 arithmetic and, as such, they remain binary. Every subblock has the same size , and the code rate is fixed as .
IiiA2 CS Encoder
For each active user , is the coded message output by the tree encoder. are mapped in to , which denote the indices of the codewords in the common codebook , where . Then the active user transmits the consecutive codewords of length , i.e., .
IiiB Decoding Process
The input to the decoder is the sum of the signals transmitted by active users plus noise after receive beamforming. The decoding process also consists of a CS decoder and tree decoder. The conventional CS decoder exploits CS techniques to recover the subblocks transmitted from all active users. The tree decoder forms code trees to piece these subblocks together to obtain the original messages.
IiiB1 CS Decoder
For each subslot s, the received signal can be expressed as
(14) 
is a row sparse matrix and can be recovered by CS techniques such as Approximate Message Passing (AMP) [donoho2009message].
For richscattering environments, an accurate and widely used statistical model for the actual channel coefficients is the Gaussian model. However, in mmWave communications, the entries cannot be approximated by a Gaussian distribution due to the lack of scatterers. Thus, we design a special activity detector for our considered scenario. Specifically, we approximate the unknown prior distribution with Gaussian mixture (GM) [wen2014channel] and EMGMAMP [vila2013expectation] models for activity detection and channel estimation. The coefficients in the th column of are approximated to be i.i.d with marginal pdf
(15) 
where is the Dirac delta, is the sparsity rate, and for the th GM component, , ,
are the weight, mean, and variance, respectively. The sparsity of the vector is captured by the sparsity rate
. The weights, means, and variances can be iteratively learned by the ExpectationMaximization (EM) algorithm.
For each subslot , the CS algorithm outputs the estimation of , i.e., . Via maximumratiocombining (MRC), the activity detector is defined as
(16) 
where is a threshold, and is expressed as
(17) 
Through the activity detector, the indices of the transmitted codewords in the common codebook are obtained and collected in the set , which is written as . As the relationship between a subblock and the corresponding codeword is a onetoone mapping, if a codeword is detected, then the corresponding subblock can be recovered automatically. The CS Decoder finally outputs the set of the subblocks and the corresponding estimated channel vectors . Notice that the index cannot represent the identity of the active user. The information that is known at the BS is that a subblock is transmitted, it comes from a certain user and the estimated channel gain of that user is . Besides, let , means the number of the subblocks collected in subslot s. is usually less than , i.e., , due to the following two reasons:

Since all users use a common codebook, the messages from different users may share some subblocks, which is defined as collision.

Due to the poor channel condition and the mistake of the CS decoder, some subblocks may be lost.
IiiB2 Tree Decoder
The traditional tree decoder in [amalladinne2020coded] aims to recover the original messages transmitted from all active users by piecing together valid sequences of the subblocks drawn from . As an initial step, the decoder fixes a subblock in as the root of a tree and gets the parity bits of the next subblock by (13). All subblocks in matching the parity bits are attached to the root. This process then moves forward. For every candidate path at stage , parity bits are computed, and the matching subblocks in are attached to this path, forming new branches. This continues until the last subslot is reached. At this point, every surviving path is output as a valid tree message.
However, the traditional tree decoder has the following two problems:

The loss of a subblock from a particular user by the CS decoder finally leads to missed detection of the original message from that user.

The parity bits to be attended in every subblock are fixed, which restricts the maximum active users that the system can serve.
To tackle the above problems, we propose two beamspace tree decoders, which are based on hard decision and soft decision, respectively. The beamspace tree decoder with hard decision has low complexity, which is suitable to the scenario of massive connectivity. The beamspace tree decoder with soft decision considers the problem of packet loss, which can be applied to the scenario with poor channel condition.
Notation  Parameter Description  
The th subblock detected in the th subslot  


The beam pattern of the subblock  





Iv Beamspace Tree Decoder with Hard Decision
The traditional tree decoder exploits the discriminating power of parity bits to stitch the subblocks together to form a valid message instead of the erroneous one. At any stage of a path during the decoding process, the subblocks meeting the parity constraints are attached to the path. Besides the valid subblock, other attached subblocks are the ones that cannot be distinguished by the parity bits. These invalid subblocks may finally lead to an erroneous message output by the tree decoder. Notice that in all subslots, the subblocks sent by different users are received by different beams at the BS according to the location of the users and scatterers. Therefore, the discriminating power of beams can be exploited to distinguish the invalid subblocks that meet the parity constraints. By leveraging the beam dimension, the decoding process can be formulated as a problem of path search in the threedimensional space, which is shown in Fig. 3.
To better describe the decoding process of the beamspace tree decoder with hard decision, Table I is given to summarize the important parameters encountered in this Section. Specifically, define the beam pattern of a subblock as a set that contains the indices of the beams that receive the subblock. Then different subblocks can be distinguished by their beam patterns. The beam pattern is written as . To get accurate beam patterns, assume the gains of the active beams obey a prior known Gaussian distribution, i.e., , where an ”active” beam means that at least the signal from one user is received by the beam. And the gains of the inactive beams obey another known Gaussian distribution, i.e., , where and . For the gains of the inactive beams, as no signal is received or the signal experiences deep fading, is close to zero. For the gains of the active beams, is large as the signals experience random fading. Using these two prior distributions, the gains of the beams can be grouped into two classes. And for each class, the mean and variance of the samples are calculated and the prior distributions can be updated, i.e., , , and . Then according to these updated distributions, we can give the decision rules of the beam patterns. Specifically, is obtained by , which is expressed as
(18) 
where .
For this proposed beamspace tree decoder, take the decoding process of a certain user for example. At the first stage, a code tree is created and a detected subblock in the first subslot becomes the root of the tree and forms the first path. The root subblock is written as and its beam pattern is . At later stages, the subblocks that meet the parity and the beam pattern matching constraints are kept. By meeting the parity constraints, is written as
(19) 
And is obtained by
(20) 
After beam pattern matching, only the subblocks in are survived. can be written as
(21) 
where . Also, means that the subblocks and are received by at least one same beam at the BS, then and have the probability to be transmitted by the same user. By beam pattern matching, the proposed beamspace tree decoder can reduce the number of surviving subblocks in each subslot, which improves the discriminating power of the decoder. A practical pruning process of this algorithm is shown in Fig. 4. The subblock is received by several beams at the BS, which is shown in the beam pattern. For every candidate path at stage , there are candidate subblocks in the common codebook that meet the parity constraints according to the parity bits. A part of them are inactive, while another part of them are discriminated by the beam pattern matching. As shown in Fig .4, the lines of the 2nd and the th candidate subblocks change from solid lines to dashed lines, which means that they are deleted, as there is no overlap between their beam patterns and the beam pattern of the root subblock at stage 1.
Finally, the proposed beamspace tree decoder with hard decision is summarized and given in Algorithm 1. is a set that contains the beam patterns of the subblocks detected in subslot s, where is expressed as .
V Beamspace Tree Decoder with Soft Decision
As described above, the CS decoder cannot always detect all the transmitted subblocks because the received signals may experience deep fading. The loss of a subblock by the CS decoder in any subslot finally leads to missed detection of the original message. This is because the traditional tree decoder and the beamspace tree decoder with hard decision just stitch the subblocks drawn from the output of the CS decoder. At any stage, according to the parity bits, the set of candidate subblocks can be obtained. At stage s, the tree decoder keeps the intersection between the candidate set and . In this proposed algorithm, we keep all the candidate subblocks and calculate the LLR values of them by implementing the MPA algorithm, which denotes the probability of whether the subblocks are transmitted. Then, we define a path metric to calculate the reliability of the consecutive subblocks and keep some reliable paths at every stage. Even if a subblock in a subslot is missed, it is possible for the path to be reliable because the path metric measures the reliability of the entire path. Therefore, the purpose of packet loss recovery is achieved.
Specifically, take a user’s decoding process for example. At stage for the th path, the number of the candidate subblocks is , and these subblocks are collected in the set . is expressed as
(22) 
where
(23) 
The difference between and in (20) is that the candidate subblocks in are drawn according to the parity bits only, thus . To reduce interference, only the received signals of those candidate subblocks are kept, which is denoted by . is written as
(24) 
where means that is in the set instead of . Then a factor graph is formed, taking the corresponding codewords as variable nodes and beam resources as resource nodes. Let and denote the number of variable nodes and resource nodes. To exploit the beam division property, the active beams in the beam pattern of the root subblock form the resource nodes. Then the remaining received signal is defined as . A certain row of comes from the th row of , where . For the sake of simplicity, define as the received signal at the th resource, as the estimated channel gain between the user transmitting the th codeword and the BS at the th resource as the th possible codeword and as a random variable that indicates whether the codeword is transmitted. Then is written as , where can be expressed as
(25) 
where
(26) 
and . To reduce the computational complexity, we resort to Gaussian Approximation (GA) as in [idma], and approximate as a complex Gaussiandistributed random variable with mean and variance , i..e., . and can be expressed as
(27)  
respectively, where
(28) 
is the log likelihood ratio (LLR) delivered from the th variable node to the th resource node. Also, denotes the LLR delivered from the th resource node to the th variable node, which is written as
(29) 
where
(30)  
(31) 
Besides, is given as
(32) 
Finally, can be expressed as
(33) 
Denote as the new paths that are split from the th path. By implementing MPA, from path at stage s, we can obtain the LLRs of the candidate subblocks in , which are written as . Learning from the way of list decoding [tal2015list], we define a path metric to calculate the reliability of the new branches from path at stage . Specifically, the PM of the new branch is written as
(34)  
where denotes the LLR of the subblock at stage in the path . The decoder calculates the PM of all the branches from every candidate path and keep some reliable paths at every stage. And at the last stage, the decoder outputs the most reliable path as the recovered message.
However, this scheme is not suitable in the case that collision occurs. As mentioned above, active users select codewords from a common codebook . Even if the dimension of is large, collisions may still occur. If the traditional tree decoder fixes a subblock transmitted by two active users at the first stage, then the decoder finally outputs two valid tree messages. According to [fengler2020pilot], we give as the average number of collisions of users on consistent s subblocks started from the first one, which is written as
(35) 
The collision of more than two users is ignored because the number is usually much smaller than 1. As grows, the collision can be ignored when . In other words, it is impossible for the valid messages of the collision users to be the new branches of the same candidate path. However, when the depth of a code tree grows, the LLR of a subblock has less impact on the PM of the entire path. Therefore, the remaining paths at stage s may all come from the new branches of the most reliable path at stage , leading to missed detection. Actually, there is no need to keep all new branches from a candidate path because the valid messages of collision users come from different candidate paths. Denote as the number of splitting paths, which means that only new branches from a candidate path are kept according to the PM. Then for the current stage, keep most reliable paths, where . This pruning process is shown in Fig. 5. The number of candidate subblocks at stage is , which is equivalent to the process in Fig. 4. At stage , for the new branches of a path, a factor graph is created to implement the MPA algorithm and give the candidate subblocks LLRs. Then the most reliable paths are kept, and others are deleted, which is shown in Fig. 5 that the lines of those deleted subblocks change from solid to dashed. Finally at stage , for the paths, the most reliable paths are kept and others are deleted. is chosen according to the tradeoff between the computational complexity and decoding performance of the decoder.
At the last stage, the number of messages output by the decoder cannot be determined because whether a collision occurs is unknown to the decoder. Notice that the traditional tree decoder outputs all the paths meeting the parity constraints as valid messages. When a collision occurs, the traditional tree decoder outputs several paths as the recovered messages. Besides, more parity bits are pushed towards later stages to reduce the probability of error according to [amalladinne2020coded]. Thus, we exploit the discriminating power of the parity bits at later stages to output the results of the beamspace tree decoder with soft decision. To summarize, list decoding is implemented at the former stages to keep the missed subblocks, and the subblocks that are full of parity bits are exploited to prune the erroneous paths at the latter stages.
However, due to the loss packet recovery, the missed detection rate decreases while the false alarm rate increases. The reason is that some undetected subblocks may be kept in the decoding process. Some of them are not transmitted actually, which may lead to false alarm. As the valid messages from collision users are not similar with each other, the invalid messages can be discriminated. Specifically, define a similarity metric , which is denoted by
(36) 
where is the indicator function, and are two messages output by a code tree. Calculate every pair of the outputs from a code tree, if , , then keep the more reliable one according to the PM, otherwise keep both, where is a threshold. By leveraging the similarity metric, the invalid messages meeting the parity constraints are deleted. The beamspace tree decoder with soft decision is summarized and given in Algorithm 2.
Vi Performance Analysis
The performance of our proposed URA system is connected with the reliability of CS techniques in each subslot and the efficiency of message stitching across different subslots. In the remainder of this section, we ignore the collision that different active users share a subblock in the first subslot.
Take the decoding process of user for example. Let be the probability that at least one subblock of user is not output by the CS decoder, be the probability of error, be the probability of missed detection, and be the probability of false alarm. is written as
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