Channel Estimation For The Multiuser Multipanel Massive Multiple Input Multiple Output System


 By increasing the number of antennas, the size and weight of massive Multiple Input Multiple Output (MIMO) become much larger and heavier, respectively. To deal with these problems, the 3GPP standardization group captured the multipanel in New Radio communications. Channel estimation is needed for accurate uplink signal recovery and downlink precoding of massive MIMO. Due to the use of hybrid structure and the nonuniform distribution of the antenna arrays, the conventional channel estimation methods cannot be used in multipanel massive MIMO. In this paper, we propose the channel estimation for the millimeter-wave multipanel MIMO systems in multiuser environments, in which the channel estimation problem is transformed to an angular domain block sparse signal recovery problem. In the following, we solve the obtained sparse model using the proposed generalized block orthogonal matching pursuit method. Also, the proper pilot sequence is designed in the proposed method. Finally, we check the accuracy of the proposed estimation algorithm, which simulation results show the superior performance of the proposed method compared to that of the classic method.


Fig. 1 Example of Uniform and Nonuniform multipanel array
Another important issue to be considered is channel estimation. Obtaining accurate channel state information (CSI) can help the receiver provide better recovery for the signal sent into the wireless environment [19][20][21]. Due to hybrid structure and nonuniform antenna array, existing channel estimation cannot be directly applied to multipanel massive MIMO [22]. There are a few kinds of research on this issue. In [22], the channel estimation and hybrid precoding are investigated for multipanel massive MIMO, in which the uplink channel is estimated using sparse modeling of the mmWave Massive MIMO channel in the angular domain to overcome in pilot overhead problem. The same method is used in [23], in which the orthogonal projection method is introduced for solving the obtained sparse model. Both these methods assume that only one user is presented in the channel estimation stage.
In this paper, we propose channel estimation of multipanel massive MIMO in the multiuser scenario. The proposed method is an extension of the method presented in [23] to the multiuser case. In the proposed method, using the angular domain sparsity feature of mmWave massive MIMO channel, the multiuser channel estimation problem is formed as a linear sparse model, in which unknown sparse vector has block sparse property so that user related elements are arranged sequentially. By solving this sparse problem, the channels of all users are estimated, simultaneously, where the generalized block Orthogonal Matching Pursuit (OMP) method is proposed for solving this sparse model. Also, the pilot design of the proposed method is investigated.
The rest of this paper is organized as follows. In the next section, the system modeling is presented. In section 3, the proposed method is introduced, and the simulation results are then presented in section 4, finally, the paper is concluded in section 5.

| SYSTEM MODELING
We consider a system in the uplink with a cell consisting of a BS with panels, where each panel is connected to only one RF chain. Also, we consider antennas in each panel; so, we need phase shifter. Thus, the total number of antennas is = . Also, we assume that there are L users in the cell, each has one antenna. In this case, the existing channel matrix between the np-th antenna panel and the l-th user is as follows [23]:

=1
(1) where, the , is the angle of arrival (AoA) of the l-th user' signal in the k-th path, respectively, K is the number of paths, ( , ) is the steering vector at the BS, which is expressed as: Also, , , ≜ , ) is the complex coefficient of the k-th path between the np-th panel and the l-th user, λ is the wavelength, is the distance between the reference panel and the np-th panel, is the phase ambiguity among the different panels, and , ∈ (−1,1] is the cosine of the AoA on the BS corresponding to the k-th path. By selecting the first panel as the reference panel, we have 1 = 0, and assuming the same distance between different panels, we have = 2 . Now we can write (1) in the matrix form as: where, In the uplink transmitting of the multipanel massive MIMO, it is assumed that a part of the receiver's beamforming process is done in the analog form; since the number of RF chains is limited. Combining the signals of a panel with the coefficients corresponding to the analog phase shifters introduces a scalar value in an RF chain. Hence, from a mathematical view, the output of the RF chain connected to the np-th panel is equivalent to the product of the received signal vector (in the np-th panel antennas) at vector , which is the phase shifter vector for Na antennas at the np-th panel, and it is expressed as [23]: where , is an analog phase shifter from the na-th antenna at the np-th panel. The value of this phase shifter is selected randomly with the uniform distribution in [0,2π). If sl be the pilot signal sent by the l-th user, the received signal in the output of the np-th panel can be described as: where ∼ ( ×1 , 2 ) be the noise vector in the antennas of np-th panel. By combining the output signals of all panels, we have ̄= ∑ + =1 (8) where ∈ ℂ ×1 is the channel vector between the BS and the l-th user as = [ 1, , 2, , ⋯ , , ] , ∈ ℂ × is the combining matrix used at the BS, which is represented as:
In the multiuser massive MIMO channel estimation, we estimate channel matrices 1 , 2 , . . , in (8), in which it is assumed that ̄, , and are known matrices or vectors. According to (8), we have a system of linear equations, where the number of unknown parameters is , and the number of equations is . As a result, sending the pilot signal at only one symbol is impossible for solving this problem. To make the problem to be solvable, the pilot sequence must be sent in many symbols (at least symbols), and by combining the obtained information, the channel matrices will be estimated. Thus, for accurate estimation, we need to assign many symbols to the pilot symbols. This leads to the pilot overhead problem. To deal with this problem. in this paper, we use compressed sensing theory for estimating the channel matrices of all users simultaneously.

| PROPOSED METHOD
In this section, we propose the compressed sensing based modeling of the multiuser multipanel channel estimation of the massive MIMO system and the method for solving it.

| Proposed sparse modeling
In practice, the number of paths associated with each user (K) is very smaller than the number of BS antennas, thus we can assume that the channel vector is sparse in the angular domain, and , can be sparsely represented as where , = [ 1, , 2, , , = (−1 + 2 ) is the i-th oversample angle, and is the oversampling factor that is selected so that, , is approximately equal to one of oversample angle values , = 1,2, . . , . In this case, , is × 1 sparse vector with K nonzero elements associated with 1, , 2, , . . , , where if = , , then , , = , , . If we consider support set of , as ℌ , = { , } ≜ { : | , , | > 0}, we can assume that ℌ 1, = ℌ 2, = ⋯ = ℌ , , which indicates that the support set of , associated with different antenna panels are equal. Since the distance between adjacent antennas in the BS antenna array is very smaller than the distance between BS and users, we can assume that the signal is received to all panels with the same AOA, which leads to similar sparsity pattern for all panels. In this case, if we represent the pilot sent by the l-th user at the n-th symbol by ( ), the received signal of the np -th panel at the output of the RF chain is equal to: Now, by combining the signals at the output of the RF chains, we have: where Thus, the is a × 1 sparse vector with nonzero elements. Since all of 1, , 2, , . . . , , have a similar support set, we rearrange the elements of so that, the nonzero elements associated with all panels become adjacent, which leads to a block sparse matrix. In other words, the vector is rewritten as the vector as = [ 1,1, , 1,2, , … , 1, , , 2,1, , 2,2, , … , 2, , , … , ,1, , ,2, , . . . , , , ] which is written at the bottom of the page. In this case, according to (14), the matrix should be changed to the matrix as. (17).
Hence, (14) is rewritten as: Now, by combining the pilot sequence of the different users, (18) is represented as: where By using the matrix relations presented at [12], we can change (20) to the vector form as follows: ] is a × 1 unknown vector that the l-th block of this vector corresponds to the l-th user. Furthermore, has a block sparsity nature with nonzero elements corresponding to actual AOA angles of paths, thus, by estimating based on (21), the channels of all users are obtained.

| Proposed generalized block OMP method
In this section, we investigate a mechanism to recover the in (21), for this purpose, the generalized version of the block OMP method is used. In (21), is an × 1 vector that is reconstructed by concatenation of = blocks which each block has elements as: which the elements are indicated in (23).
In the following, we explain each step of the block OMP algorithm, named algorithm 1. After a selection of the initial values in step 1, we calculate the multiplication of the matrix D and the residual vector r in step 2. In step 3, we compute the matrix , which its j-th element is the energy of j-th blocks of vector p, then the index of a block with the largest energy ( * ) is calculated in step 4, and corresponding indices in p is named as . This finally leads to the detection of the location of the active elements. The new active elements' location is added with that of the previous in stage 5. In the next stage, the nonzero elements of sparse vector are estimated with LS criterion by using the subset of D associated with the active elements' locations. Finally, at step 7, the residual vector is updated by removing the portion of the selected columns in the previous steps.

3 | Pilot sequence design
For estimating the channels between antennas of BS and all users, each user must transmit an individual pilot sequence. To reduce the pilot overhead problem, we assume that all users simultaneously transmit their pilot sequences and all channels are estimated using a compressive sensing theory. In this section, we discuss the constrain of the pilot sequences of each user and analog phase shifter values to achieve the accurate estimation of the multipanel MIMO channels based on the block sparsity algorithm investigated in the previous section.
For solving a sparse problem using compressive sensing theory, mutual coherence has an important role in the accuracy of estimation. This parameter defines as follow [26] = 1 ≠ 2 In which indicates the i-th column of D. According to (24) we have 0 ≤ ≤ 1. For the small mutual coherence, the compressed sensing algorithms have better performance. In the case that the column of D has strong similarity, the value of mutual coherence is large which leads to low accuracy estimation.
Thus, to achieve high accuracy channel estimation, we need to have a small value for . According to = ⊗ , the Kronecker multiplication between the pilot sequences matrix and the measurement matrix creates a challenge to achieve the minimum value of the mutual coherence parameter. To better understand this issue, we display D as follow: As you can see, the values of the pilot sequence play a key role in differentiating some columns of D. For example, according to (17) and (25), the pilot sequences of 1-th and 2th user determinate the distinction between the 1-th and ( + 1)-th columns of D. Therefore, to minimize the mutual coherence, addition to that the pilot sequence have different values in different symbols, each user must have a unique pilot sequence. Also, to ensure unit power constraint for transmitted pilot sequence, we assume that each element of the pilot has unit magnitude. To meet these constraints, we choose the n-th pilot of l-th user as s l (n) = , in which is a random variable with uniform distribution over [0,2 ].
On the other hand, according to (17) and (25), the mutual coherence of some adjacent column of D (such as 1-th to -th or ( + 1)-th to 2 -th columns) depends on the values of analog beamforming coefficients. If the phase shifter vectors associated with all panels or all training symbols are equal, a large value is obtained for the mutual coherence of these adjacent columns. Thus, to deal with this problem, we select different values for the phase shifter vectors of all panels and all training symbols.

| SIMULATION RESULTS
In this section, our goal is to evaluate the performance of the proposed algorithm in the multiuser scenario, using the Normalized Mean Square Error (NMSE) and the Bit Error Rate (BER) metrics, and compare them with the results obtained by the classical method. The classical method refers to the algorithm presented in [23] is proposed in the single-user scenario. To create the possibility to use the results of this algorithm for comparison with the multiuser scenario, we assume that each user sends a pilot sequence in certain symbols, individually, then its channel is estimated using the method of [23]. However, in the proposed method, we simultaneously estimate the channel matrix of all users and do not discriminate among the users in the estimation process. In the figures of simulation results, we legend the results obtained from the proposed algorithm as "proposed", and the results obtained from the classical algorithm as "classic". Also, these results have been calculated using computer simulations in the MATLAB environment based on the Monte Carlo technique, averaging over the 5000 trials for NMSE criterion and the 50000 trials for BER criterion.
In the simulations, the number of active paths between each user and the BS has been selected as k =2, which indicates a very lower number of paths. The reason is the high fading of the signals from the different paths, due to the inherent losses of the mmWave signals. The number of antennas within each panel is 32, and the number of panels is 4, which leads to the total number of used antennas be 128.

| Comparing the methods via NMSE
The first criterion for comparing different estimation methods is NMSE which is defined as: (26) in which and ̂ are the actual and estimated channel vector of l-th user. In the first simulation, we select L=2, Ks = 8, and Nφ = 8, and compare the NMSE versus SNR for the proposed and the classical algorithms. The results are shown in Fig. 2, indicate that as the SNR increases, the NMSE decreases to an acceptable level, as expected. At low SNRs, both methods have a relatively similar performance; however, by increasing the SNR, the proposed method presents more favorable results. For example, at SNR=5 dB, the results obtained for the proposed algorithm and the classical algorithm are 0.2247 and 0.2468, respectively. However, at SNR=35 dB, the values obtained for the proposed algorithm and the classical algorithm are 0.0991 and 0.1365, respectively. In addition, we see an error floor in the classical algorithm at SNR=30 dB, and the NMSE performance no longer improves by increasing the SNR. While in the proposed algorithm, there is no such an error floor, and the NMSE performance is improving by increasing the SNR. Therefore, to investigate the effect of pilot sequence length, we assume Nφ = 6 in the second simulation. In this simulation, we still have Ks = 8, and the NMSE performances versus the SNR are provided. The results obtained in Fig 3 indicate that by decreasing the length of the pilot sequence, the channel estimation performance decreases to some extent. This is because the measurement matrix M finds fewer rows than the case in Fig 2, which increases the error compared to the case with Nφ = 8. Therefore, the channel estimation accuracy is reduced in Fig 3 compared to that of Fig 2. This reduction in accuracy is lower in the proposed algorithm than the classic algorithm. For example, at SNR=5 dB, the NMSE obtained for the proposed and classic methods are 0.2996 and 0.3494, respectively, and for SNR=35 dB, it is 0.1733 and 0.2472, respectively. By comparing these results with that obtained in the case Nφ = 8, we see that the differences between the NMSE of Nφ = 6 and Nφ = 8 cases at SNR=5 dB are 4.98% and 2.21%, respectively, and at SNR=35dB, these differences are 7.39% and 3.74%, respectively. This means that by reducing the amount of Nφ from 8 to 6, the accuracy of the classical method reduced twice as much as the proposed method.  Another challenge to consider is the change in the number of users and its impact on the channel estimation process. To obtain this issue, by establishing conditions similar to Fig 2, we increase the user number to 4, which the simulation result is shown in Fig 6. As seen, the result obtained for the classical algorithm is consistent with the result observed in Fig 2. But in the form of the proposed algorithm, we see an accuracy increase. This is due to varying the received paths by increasing the number of users and thus reducing the mutual coherence effect. In the following, we will examine the extent of changes in other parameters and their impact on achieving more desirable quality. Therefore, by assuming SNR=20dB and changing the other parameters, namely Np (number of panels) and Nφ (number of pilot symbols), we calculate the NMSE value.
As seen in Fig 7, increasing the Np reduces the NMSE in both the proposed and the classical algorithms. This is due to the increasing dimension of the measurement matrix, which leads to a higher accurate sparse model solving. Another point regarding the results obtained in Fig 7 is that the proposed algorithm is better than the classical algorithm.
Therefore, by increasing the number of panels the higher accurate channel estimation is obtained in the multipanel massive MIMO system. However, the main point is the increase in the costs, which increases with increasing the number of panels. Therefore, to achieve an acceptable accuracy and reasonable implementation costs in a multipanel MIMO system, we need to trade-off between the costs and the estimation accuracy to find the optimal number of panels. In Fig. 8, by assuming SNR=20 dB, we compare the effect of the pilot sequence length changing on the accuracy of channel estimation. As seen, by increasing the Nφ, the NMSE decreases for both methods, which is provided at the cost of reducing the throughput of the system by increasing the pilot overhead. The measurement matrix is a factor in increasing this accuracy. Also, in Fig 8, we see that the proposed method is better than the classical method.

| Comparing the methods via BER
In the following, we use the BER criterion to evaluate the quality of channel estimation. In this case, the important point raised in the detection step is the use of the estimated channel matrix instead of the actual channel matrix. For detecting the users' signals, we use the Maximum Likelihood (ML) criterion, in which ( ) is estimated usning the following method: In which ( ) is the trasmtted signals vactor that its l-th element indicates the 4PSK modulator output signals of the l-th user in the n-th symbol, ̃( ) is the received signal vector that its n-th element indicates the output signal of n-th pannel's RF chain in the n-th symbol, and = [ 1 , 2 , . . . , ] with =̂ , ̂= [̂, 1 ,̂, 2 , . . . ,̂, ].
In the next simulation, we use the estimated channel matrix is obtained by parameters Ks = 4, Nφ = 8, and L=2 in (27), and compare the BER versus SNR for uplink transmitting the 2 users' signal. The result that is shown in Fig. 9 indicates that by increasing SNR, the BER decreases, which is as expected. Also, the BER associated with the proposed method is lower than that of the classic method. Since the channel estimation accuracy of the proposed method is better than that of the classic method, the detection accuracy of the proposed method is higher than that of the classic method.

| CONCLUSION
In this paper, we proposed a channel estimation method for the millimeter wave multiuser multipanel MIMO systems. In the proposed method the channel estimation problem transformed to an angular domain block sparse signal recovery problem, which was solved by the proposed generalized block OMP method. The accuracy of the proposed method was measured using NMSE and BER criteria, which simulation results indicated the superior performance of the proposed method compared to that of the classic method.