Nowadays, the number of wireless connected devices has been vastly increased by the development of applications such as Internet of Things (IoT), machine type communications, social networking and next generations of cellular communications. Therefore, massive multiple-input multiple-output (MIMO) systems have received remarkable attention during the past few years. The 5G wireless technology has identified three main applications equipped with this capability : massive machine-type communications (mMTC), enhanced mobile broadband communications (eMBB) and ultra-reliable low-latency communications (URLLC). There are lots of advantages with massive MIMO systems such as increasing the system throughput and the energy efficiency [1, 2]. However, all of them are dependent on accurate channel state information (CSI) in coherent transmission which is a highly challenging task. In massive MIMO system, deploying the Frequency Division Duplexing (FDD) mode is inefficient and Time division duplexing (TDD) mode is often used due to the fact that the reciprocity of channel estimation is applicable. However, it requires a portion of resources as pilots or training signal in each Coherence Interval (CI) to estimate channels corresponding to active users and it is directly proportional to the number of users. This is a big challenge and limitation for massive access  which prevents to admit massive connected users. This challenge is more enhanced in mMTC and crowded scenario of MBB (cMBB) in which the number of users are more than the length of orthogonal pilot sequences and data transmission from a massive number of devices in a dynamic and timely manner is expected. Specifically, the BS must determine how to dedicate pilots and resources to users in order to send their data. There are two general strategies for user access in such scenarios: grant-based and grant-free. In the grant-based random access (RA), each user equipment (UE) selects a random preamble from a predefined orthogonal pool and sends the same to BS. If there is no pilot contamination, the BS identifies active UEs by this preamble and transmits a message to active users. Then, each active user sends a connection request to BS in order to ask for resources to be granted for data transmission. In this method, there might exist contentions in connection requests of multiple UEs when they exploit the same preamble. In this case, either the BS resolves the contention or resets the RA procedure. Since large number of collisions occurs, the BS cannot resolve all of the contentions, thus, many users are not able to access the network. Another category of RA is grant-free in which each active user simultaneously transmits both pilot sequences and data. The BS exploits the pilots to identify the active users and to estimate the corresponding channels and then decodes their data sequence [3, 4, 1]. Both methods become challenging in case of massive users e.g. in crowded communication scenarios as the coherence time is limited and thus few orthogonal pilots are available to assign a large number of users. Consequently, most of the users can not access the network. Besides, if there are some RA methods to admit maximum number of active users, half of the system resources should be used as overhead [5, 1] preventing to have higher spectral efficiency.
To solve these challenges, it is beneficial to leverage the intrinsic features of massive MIMO channels (angular domain sparsity of massive MIMO channels) as well as the sporadic traffic of users to detect the active users and estimate their spatial features to define and apply new protocols of random access in crowded scenarios of massive MIMO systems. The task of extracting such features from a few number of measurements builds upon a well-known framework called compressed sensing (CS) . Precisely, CS suggests a framework for recovering discrete-index parameters i.e. the unknown parameters are confined to be on a predefined domain of grids. There is also a more recent framework called continuous compressed sensing ( or super-resolution) [7, 8, 9, 10] which assumes that the unknown parameters can lie anywhere and are not confined to be on the predefined grids.
These simple features are available in massive MIMO systems with massive number of users. For example, the physical channel of massive MIMO systems, employed in high frequencies (millimeter wave), have continuous sparse structure i.e. signal is received out of few off-grid (continuous) angles in BS antenna arrays and they can have any arbitrary values . This is due to the fact that signals with higher frequencies are more likely to be blocked by obstructions and few multi-path components (MPCs) contribute to the channel. Another feature is sporadic traffic of massive users which means only few users want to send their data at the same time. It is also been shown that the channels between users and BS exhibit a clustered continuous sparsity pattern . This means that the Angles of Arrival (AoA) corresponding to each user are alongside each other, few number of clusters are active and the AoAs are continuous-index values.
This work revolves around this issue. Specifically, these features are leveraged up to their utmost levels to simultaneously estimate the channels and active users’ data. For this task, a novel framework is proposed that takes the two vector unknowns i.e. channels vector and UEs data vector to a matrix variable lying in a higher dimensional space. Then, a novel optimization problem is proposed to recover the matrix variable exactly. Next, a clustering-based algorithm is proposed to detach the angles corresponding to each user. Lastly, an alternative optimization algorithm is developed to estimate complex amplitudes of channels and data/primary pilots transmitted by active users ina blind way. Our approach can also be employed in the process of random access to pilots (RAP) in cMBB/eMBB and data recovery in mMTC. By our approach, the limitation in the number of adopted simultaneous users in crowded scenarios of massive MIMO systems would be resolved.
I-a Prior works and Key Differences
In recent years, there are many works regarding RA methods such as [13, 12, 14, 15]. In , the authors suggest a strongest-user collision resolution protocol to resolve the limitation of admitted active users in cMBB. The number of active users in their method is bound to the number of orthogonal pilots.
In , the authors propose Approximate Message Passing (AMP)-based method called BiG-AMP to recover both channels and data simultaneously. Their approach needs prior distributions of channels and data. Apart from that, their method does not consider the inherent clustered continuous sparsity of massive MIMO channels in crowded scenarios. Thus, high performance might not be achieved.
In [12, 14], the authors propose active user detection and channel estimation scheme based on CS.  designed an Approximate Message Passing (AMP) algorithm to detect active users as well as to recover channels based on the transmitted pilots by users.  uses orthogonal matching pursuit (OMP) to perform channel estimation. Besides the fact the AoAs in these works are assumed to be on a predefined domain of grids, their strategy needs known pilot sequences in advance which substantially increases system overhead and prevents from reaching a high spectral efficiency.
I-B Outline and Notations
In Section II, the system model of massive MIMO is presented. Section III is about our proposed blind super resolution method and provides an algorithm for blind detection and channel estimation. Lastly, Section IV provides some numerical experiments to verify our proposed method. Lastly, the paper is concluded in Section V.
We use boldface lower-and upper-case letters for vectors and matrices, respectively. The -th element of a vector e.g. and the element of a matrix e.g. are respectively shown by and . For vector and matrix , the norm and Frobenius norm are defined respectively as . stands for the sign function of which is applied component-wise and separately to the real and imaginary parts. means that is a positive semidefinite matrix. For two arbitrary matrices , represents the trace of which is denoted by . is a operator transforming an arbitrary matrix to a reduced matrix with rows indexed by .
Ii System Model
In this section, we first introduce the signal model in uplink access for MIMO-OFDM systems [12, Section II]. Then, we provide the intrinsic features of channel matrix in these systems. We consider a well-known scenario where there is one BS equipped with an -element uniform linear array (ULA) and single antenna users. The channel between the -th user and the BS is modeled as (see [12, Equ. 2] or [17, Equ. 7]):
in which is the number of physical paths between -th user and BS, is the Angle of Arrival (AoA) of the -th path, is the complex gain of the -th path,
is the receive steering vector, , and . Due the sparse characteristics of massive MIMO channels, it holds that . The channel is considered to be block fading, i.e., it is constant during several CIs where each is denoted by (see Figure 1). The received signal at the BS after time slots at the sensors indexed by (with length ) becomes in the form of (, [12, Equ. 3] or [17, Equ. 1]):
where is the transmitted signal from -th UE, is the additive noise matrix, each element of which is distributed as and . Inspired by , and by defining and , can be expressed as a sparse linear combinations of the matrix atoms in the atomic set which are regarded as building blocks of .
Iii Proposed Blind Super-resolution method
In (3), we have an under-determined set of equations with observations and unknowns. While this problem has infinite number of solutions, it could be transformed to a tractable problem by assuming that which is reasonable in massive MIMO systems. This strategy is built upon well-known continuous CS approaches [7, 8, 20] and provides a unique optimal set of solutions for matrices s in (3) leading to the least number of atoms under the affine constraints of (3). Thus, we form the following optimization problem to reflect the structure of s:
where is the atomic function which computes the least number of atoms to describe . As (4) is an NP-hard problem in general, we relax it into a convex optimization problem which is stated as:
where the atomic norm is the best convex surrogate for atomic function and is defined as
where is the convex hull of .
One way for identifying the AoAs (which we used in our simulations) is by leveraging the solution of the dual problem of (5) which is provided below:
is the adjoint operator of . Then, we use the following lemma (adapted from [10, Lemma 1]) which guarantees the uniqueness of the solution in the noiseless case:
Denote the set of AoAs from -th users by . The solutions of obtained from (5) in the noiseless case () are unique if there exist dual matrices such that the vector-valued dual polynomials satisfy the conditions
This lemma shows that the AoAs can be easily estimated by identifying locations where achieves . As stated in , this provides a good insight about the procedure of finding AoAs in the noisy case. Specifically, we find the AoAs by identifying the ones that .
After obtaining the estimated AOAs, a clustering-based algorithm  is employed to detect the AOAs of the active users denoted by . Here, is the estimated indices of active users with length . By knowing the DOAs corresponding to each active user, (3) turns into the following equation:
where is the steering matrix of -th active user and . The task of recovering the unknown matrices and from is a bi-linear inverse problem. For this task, we propose an alternating optimization to jointly estimate complex channel coefficients and transmitted data corresponding to active users. First, we begin with a random distributed on the unit sphere. By replacing in (9), we deal with the following least square problem:
which by considering and orthogonal s, has the solution
By integrating the latter expression into (9), we must solve the following least square optimization: where and . The latter optimization has also the closed-form solution
Finally, the steps (11) and (12) are alternatively performed to yield the final solution.The pseudo code of the proposed method which is indeed a summary of the aforementioned steps is provided in Algorithm 1.
Select a uniformly distributed random vector for transmitted data as2: Solve the dual problem (7) to obtain as follows: Obtain the dual polynomial . Localize the estimated angle by the following two methods: Discretize on a fine grid up to a desired accuracy and find and by identifying locations where achieves to according to Lemma 1. The total number of angles reaching specifies an estimate for the total number of MPCs i.e. 2: 3:
Apply the k-means methods to cluster the angles of channel UEs.4: 5: for to do 6: Identify the corresponding indices with the -th label. 7: Estimate the length of -th cluster i.e. . 8: Obtain the angles corresponding to the -th cluster (UE) i.e. 9: Estimate the steering matrix as 10: 11: end for 12: for to maxiter do 13: Perform channel estimation to recover according to (10). 14: Perform channel estimation to recover according to (12). 15: . 16: end for Return: .
Our algorithm provides a grant-free scheme which simultaneously estimates data as well as channels. The only assumption on the data to be unambiguously recovered is positivity i.e. and normalized power for all active users . The proposed method has implications for data recovery in mMTC and primary pilot and AOA detection for RAP in cMBB. For example, CI in cMBB divides into 2 parts: RAP and coherent transmission blocks (see Figure 1). By utilizing this novel approach in the RAP block, active users detection and AoA estimation are performed by BS [14, Section 5]. By our method, BS can identify many users at the same time with their AoAs and with the lowest level of spending system resources. In fact, the number of active users that can access the network depends on the complexity that BS can bear. There is no need for orthogonality of primary pilots needed for RAP process in  and prior distribution of pilots and channels as is the case in AMP-based approaches [17, 12]. Subsequently, in the coherent transmission step, by knowing the exact AoAs of active users, BS can easily allocate orthogonal pilots to non-overlapped UEs and estimate their corresponding data via (12).
In this section, we perform some numerical experiments to evaluate the performance of our proposed algorithm in blind channel and data reconstruction. We use SDPT3 package of CVX  in MATLAB for solving problem (7). The number of BS antennas is set to . We assume the one-ring model for the channel . The AoAs are randomly chosen from . The path amplitudes are distributed as . The separation between any receive antennas at BS are set to . The observed sensors at BS () are randomly chosen out of . Also, the maximum number of iterations in Algorithm 1 denoted by maxiter is fixed to . The upper bound of noise variance is chosen as . The signal to noise ratio is defined by , the probability of detection enhances by increasing the number
. The upper bound of noise variance is chosen as
. The signal to noise ratio is defined by. First, in Figure 2, we show the successful procedure of active user detection and clustering with parameters , , , , . The maximum number of MPCs is fixed to . Figure 2 shows norm of the vector dual polynomial at different angles in terms of radian. The estimated angles are found by identifying locations that . The number of peaks provides an estimate for . After finding the angles, we apply k-means method to cluster the angles corresponding to active users. The number of elements inside each cluster provides an estimate for . Then, steps 12 to 16 of Algorithm 1 are employed to obtain the pilots and channels. The performance of our algorithm in recovering users’ data, channel amplitudes and AoAs is evaluated using normalized mean square error (NMSE) respectively defined by , and . The Monte-Carlo iterations to approximate the expectation is set to . The evaluation for the first experiment are as follows: . In the second experiment, we evaluate the performance of Algorithm 1 in different noise values in a more practical scenario with parameters . As it turns out from the bottom image of Figure 3, NMSEs tends to zero at high SNRs which in turn implies that our proposed method performs well in estimating users’ data, complex channel amplitudes and AoAs of active users. In the experiment shown in the Top-right image of Figure 3, we compare our method with  for different number of antennas. For both methods, we obtain NMSE of the channel matrix defined by . As it can be observed, our blind method performs better in channel estimation than  which assumes the users’ data known. In the last experiment, we evaluate the active user detection performance by a detection rate criterion defined as where the numerator returns the number of differences between the true active users and the estimates. As shown in the top-left image of Figure 3
, the probability of detection enhances by increasing the numberof observed arrays at BS.
In this work, we designed a novel SDP optimization for AOA detection and channel estimation of active users and consequently developed a blind spatial-based random access solution which is applicable to crowded massive MIMO systems. Specifically, we showed that the recovery of both pilots and channels are possible via observing a few noisy measurements in blind manner. For this task, we applied a clustering method to the output of the SDP to demix the AoAs corresponding to active users. Then, an alternating-based approach is designed to recover the pilots as well as the channels. Our proposed algorithm and strategy are verified by simulation results which show its high performance.
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